Multi-fields in multiferroic materials induced by eigenfields and remote loads

This work presents the analytical solution to the multi-fields in the whole domain of 3D multiferroic materials with an ellipsoidal inclusion/inhomogeneity induced by eigenfields and remote loads. It is a unified approach for the fully-coupled analysis of single-phase multiferroic materials and multiferroic composite materials, which are verified by comparing with available results in the literature and finite element analysis. It is found that, inside the inclusion, the magnetic field due to the spontaneous polarization and the electric field due to the spontaneous magnetization increase monotonically with the aspect ratio of the inclusion for single-phase multiferroic materials, but they first increase and then decrease with the aspect ratio of the inclusion for multiferroic composite materials with much larger maximum magnitudes. Moreover, in the matrix of the piezoelectric-piezomagnetic heterostructure, some components of the multi-fields change dramatically with the aspect ratio of the inhomogeneity, but other components vary insignificantly. Furthermore, although the generalized strain field is usually not uniform in the matrix, the uniformity condition could be still achieved by tuning the eigenfields and remote loads even regardless of the specific shape of the ellipsoidal inhomogeneity. These results could be helpful for the design of multiferroic materials.