A general expression of magnetic force for soft ferromagnetic plates in complex magnetic fields

The experiments of ferromagnetic plates in different magnetic environments exhibit two distinct phenomena, i.e. the magnetoelastic instability of a ferromagnetic plate in transverse magnetic fields, and the increase of natural frequency of a ferromagnetic plate with low susceptibility in an inplane magnetic field. Although these two typical phenomena can be predicted separately by two kinds of theoretical models in which the magnetic forces are formulated by totally different expressions, no theoretical model has been found to commonly describe them. This makes it difficult to predict theoretically magnetoelastic interaction of a ferromagnetic structure in complex magnetic environment. A variational principle, here, is proposed to establish the governing equations of magnetoelastic interaction for soft ferromagnetic thin plate structures under complex magnetic fields. The functional is chosen as the summation of the magnetic energy and the strain energy as well as the external work from applied magnetic fields. From manipulations of the variational principle, the governing equations of the magnetic field and mechanical deformation together with an expression of equivalent magnetic force exerted on the ferromagnetic plates are obtained. It is shown that this theoretical model can commonly characterize the experimental phenomena of the magnetoelastic interaction aforementioned.