In this paper, we are concerned with a model for the magneto-elastic interactions of a three-dimensional elastic body and a two-dimensional flexible plate, which is attached to the flat flexible part of the surface of the body. Both the solid body and the plate are permeated by magnetic fields. The mathematical model is analyzed from the point of view of existence and uniqueness and stabilization.It turns out that, in the presence of the magnetic fields in the solid and the plate, strong stabilization can be achieved under viscous damping in the plate in one direction that is determined by the nature of the primary magnetic fields in the body and the plate.
IntroductionIn this paper, we consider the interaction of a three-dimensional elastic electrically conducting solid body, which has a flat flexible horizontal base, and a thin elastic electrically conducting plate, the interface, which is mounted on the base of the solid. The remainder of the surface of the body is assumed to be rigid. Whilst it is assumed that the 3 D body is subject to in-plane and transversal deflections, we will assume that the transversal motions of the thin 2 D plate are negligibly small relative to its in-plane motions and thus omitted. This is a reasonable assumption when the body is considerably more flexible in the horizontal direction than in the vertical direction.Perfect contact between the body and the plate is assumed at the interface, so that the magneto-elastic motions of the body at its base are 'in unison' with those of the plate. This entails that the composite dynamics of the structure is governed by contact constitutive equations, which stipulate that the elastic as well as the magnetic fields that may be different in the interiors of the body and the plate match at the interface. Moreover, account is taken of the surface tractions at the interface resulting from the magneto-elastic motions of the body. When integrated into the plate dynamics, these surface tractions are manifested as additional terms in the system for the magneto-elastic plate, that in view of the contact assumption furnishes, in turn, a system of dynamical boundary conditions for the system for the solid body.Primarily our aim is to consider the model for the composite magneto-elastic structure from the point of view of stabilization, in particular, to determine the effectiveness of the magnetic dissipation in the solid and the plate. Our interest in this problem was inspired, on the one hand, by studies of the classical system of magneto-elasticity, and, on the other hand, by a flourishing interest in hybrid systems in elasticity in recent years. For the sake of coherence we mention only a few contributions-more references to the literature are provided at the end of this section. In the area of classical magneto-elastic systems, Perla Menzala and Zuazua [1] established strong stabilization of the model when the domain in a three-dimensional space is simply connected, without having to incorporate mechanical damping in the model. Further to this work M...