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On a model for the magnetoelastic vibrations of a perfectly conducting Mindlin-Timoshenko plate. (English) Zbl 1465.74072

Appl. Anal. 100, No. 2, 322-334 (2021); correction ibid. 100, No. 2, 464 (2021).
In continuation to own previous work [Z. Angew. Math. Phys. 66, No. 1, 113–128 (2015; Zbl 1317.74052)], the author investigates a model for the magnetoelastic vibrations of a perfectly conducting Mindlin-Timoshenko plate. Well-posedness of the model is established.
The author recapitulates the procedure followed in a previous work to derive the magnetoelastic model, taking into account the Hencky-Bollé approximations of the components of the elastic field, with similar assumptions of linear approximations for the components of the induced magnetic field.
The case of infinite electrical conductivity is then considered. It turns out that the model in this case reduces to a Mindlin-Timoshenko plate model augmented by a Lorentz force term involving only the bias magnetic field. Analogy is established with a model for one-dimensional nano-composites with axial magnetic field.
Existence and uniqueness properties of the model with infinite conductivity are studied within the framework of semigroup theory by reducing the initial-boundary value problem to a perturbed abstract evolution equation. A formula is provided for the time rate of change of the energy of the system.
A final section is devoted to the stabilization of the unperturbed model with infinite conductivity equipped with Kelvin-Voigt damping in the system for the shear angles of plate filaments. It is shown that the model in this case exhibits a slower rate of polynomial decay than the coupled hyperbolic-parabolic model, with no mechanical damping, for the magnetoelastic vibrations of a Mindlin-Timoshenko plate with finite conductivity. This demonstrates the damping effect of the dynamic magnetic field.

MSC:

74H45 Vibrations in dynamical problems in solid mechanics
74K20 Plates
74F15 Electromagnetic effects in solid mechanics
74H20 Existence of solutions of dynamical problems in solid mechanics
74H25 Uniqueness of solutions of dynamical problems in solid mechanics
74H30 Regularity of solutions of dynamical problems in solid mechanics

Citations:

Zbl 1317.74052
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References:

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