On a model for the magnetoelastic vibrations of a perfectly conducting Mindlin–Timoshenko plate

Abstract

This article is a continuation of our earlier work on models for the magnetoelastic interactions of an elastic field and a magnetic field in an electrically conducting Mindlin–Timoshenko plate. In the present work our focus is on the role of the electrical conductivity. It turns out that when the plate exhibits ‘infinite’ electrical conductivity, i.e. the electrical resistance is negligible compared to other effects, then the dissipative property of the energy associated with the original model is forfeited and the model reduces to a Mindlin–Timoshenko plate model augmented by a Lorentz force term. We establish well-posedness of the model without imposing the condition of negligibly small normal magnetic stresses at the plate faces in our earlier work. This entails the analysis of a perturbed evolution equation. Finally polynomial stabilization of the unperturbed model is achieved by incorporating damping of Kelvin–Voigt type only in the system for the shear angles of plate filaments.

Keywords:

• Mindlin–Timoshenko plate
• magnetic field
• infinite electrical conductivity
• perturbed evolution equation
• polynomial decay

AMS Subject Classifications:

• 74K10
• 93B15

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Correction

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1 The incompressibility condition was recently lifted by Ferreira and Muñoz Rivera [12] by introducing thermal effects into the model.

2 To the best of our knowledge the resolvent criterion developed by Borichev and Tomilov, has not been applied to perturbations of evolution equations. We are only aware of work by Hai [19] on inhomogeneous evolution equations in which the inhomogeneous term is required to be of the form $f\left(t\right)/t$.