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The current work reveals the exact nature of the induced interior and exterior coupled-fields of magneto-electro-elastic ellipsoidal inclusions with non-uniform generalized eigenfields, consisting of eigenstrain, eigenelectric, and eigenmagnetic fields. The medium has general rectilinear anisotropic elastic moduli, piezoelectric, piezomagnetic, dielectric, magneto-electric, and magnetic permeability tensors. The non-uniform eigenfields are assumed to be representable as the product of any arbitrary functions whose arguments are the equation of the boundary of the ellipsoidal inclusion with homogeneous polynomials. As a special case, it has been proved that the homogeneous polynomial eigenstrain, eigenelectric, and eigenmagnetic fields simultaneously induce inhomogeneous polynomial strain, electric, magnetic, stress, electric displacement, and magnetic induction fields of the same degree at the interior points of the inclusion. Certain series forms of the eigenfields in cylindrical coordinates are also treated. A special class of impotent eigenstrain, eigenelectric, and eigenmagnetic fields which give rise to vanishing strain, electric, and magnetic fields within the ellipsoidal domain is introduced and proved. Furthermore, the energies pertinent to the magneto-electro-elastic inclusions with arbitrary geometries and eigenfields are formulated. Also, the exact analytical expressions of the magneto-electro-elastic jump conditions of the generalized stress and the gradient of the generalized displacement fields are obtained. A number of theorems, lemmas, and corollaries in connection with the exact nature of the solution are stated and proved for the first time. The presented formulations and theoretical developments are of great value in the determination of the exact induced interior and exterior coupled-fields of anisotropic quantum wire/quantum dot structures as well as anisotropic piezoelectric/piezomagnetic composites.

Determination of the effective elastic properties of unidirectional short fiber reinforced composites (SFRCs) is a key fundamental requirement which is usually estimated by mean-field homogenization (MFH) methods. Available classical MFH methods only account for volume fraction and aspect ratio of the fibers and do not consider the physical cylindrical geometry of the fibers. Moreover, classical MFHs are not designed for predicting the effective properties of SFRCs with distinct packing configurations and order of fibers. In this work, new modified Eshelby tensors associated with an enhanced MFH are developed which resolve the mentioned limitations of the available MFH methods and preserve their computational efficiency. While classical Eshelby tensors used in MFHs depend only on the aspect ratio of the fictitious ellipsoidal inclusion and material properties of the matrix, the introduced modified Eshelby tensors additionally account for physical cylindrical geometry as well as interactions of the short fibers with different packing configurations. Moreover, non-uniform homogenizing eigenstrains required for accurate homogenization of the short fibers are also incorporated by the proposed analytical treatment. Predictions of the developed enhanced MFH for various packing configurations are compared with those obtained from Finite Element Method implementing periodic boundary conditions and very good agreements are observed. It is shown that packing configuration and geometrical specifications of short fibers can significantly affect the effective properties of the SFRC.

The exact nature of the induced coupled-fields of anisotropic magneto-electro-elastic ellipsoidal inclusions, multi-inclusions, and inhomogeneities with non-uniform eigenfields under polynomial magneto-electro-elastic far-field loadings is of particular interest. For the sake of prediction of the induced coupled-fields of magneto-electro-elastic multi-inclusions due to piecewise polynomial generalized eigenfields several theorems and corollaries are stated and proved. Some classes of impotent generalized eigenfields associated with encapsulated ellipsoidal multi-inclusion result in vanishing generalized disturbance strains within the innermost ellipsoidal domain. On the other hand, it is found that there are certain classes of impotent generalized eigenfields for which the generalized disturbance stresses vanish everywhere. Furthermore, it is proved that there exists a unique equivalent magneto-electro-elastic inclusion associated with an anisotropic magneto-electro-elastic ellipsoidal inhomogeneity subjected to polynomial far-field loadings. It is shown that associated with every anisotropic magneto-electro-elastic ellipsoidal inhomogeneity, there exists a set of infinite numbers of far-field loadings for which the inhomogeneity does not induce any disturbance generalized stresses anywhere within the entire medium. In other words, under certain far-field loadings the inhomogeneity is not sensed by the surrounding magneto-electro-elastic medium.

The dynamic equivalent inclusion method (DEIM) which was first proposed by Fu and Mura (1983), in its original context has some shortcomings, which were pointed out and remedied by Shodja and Delfani (2009) who introduced the new consistency conditions along with the related micromechanically substantiated notion of eigenstress and eigenbody-force fields. However, these theories are bound to elastic media with isotropic phases. The present work extends the idea of the above-mentioned new DEIM to the dynamic electro-mechanical equivalent inclusion method (DEMEIM) for the treatment of the scattering of SH-waves by a two-phase circular piezoelectric obstacle bonded to a third phase piezoelectric matrix. All the three transversely isotropic media have the same rotational axis of symmetry and the same poling direction which are parallel to the axis of the coated fiber, but perpendicular to the direction of propagation of the incident SH-wave. In general, the nested circular media are considered to be eccentric, i.e., the core fiber has a coating with variable thickness. Realization of the nature of the behavior of the field quantities a priori and its appropriate implementation in to the new extended consistency conditions is a critical step to insure a rigorous mathematical framework. As it will be shown, the expansion of the Green’s function and the eigenelectric, eigenstress, and eigenbody-force fields in terms of the eigenfunctions of the pertinent field equations rather than the commonly considered polynomials in the traditional equivalent inclusion method (EIM) leads to an accurate solution with high convergence rate. The exact analytical expression for the total scattering cross-section which is influenced by the piezoelectric couplings is derived. The effects of the piezoelectric couplings and the properties of the fiber, coating, and the matrix as well as the wave number on the electro-mechanical scattered fields are examined.