Coupled PDE Models for the Mindlin-Timoshenko Plate

Marié Grobbelaar^*

At the IFIP TC7 conference in Berlin (2011) a model for the magnetoelastic interactions between an electrically conducting Mindlin-Timoshenko plate and a magnetic field, that permeates the plate, was presented. When the uniform magnetic field is aligned with the mid-plane of the plate and the elastic variables satisfy Dirichlet boundary conditions, a frequency domain method yielded polynomial stability of the model, weak solutions decaying at a rate of order $t^{-\frac{1}{4}}$ when the data are sufficiently smooth. A specific feature of the model is the strong coupling between the elastic and magnetic variables, i.e. in all the PDEs of the model. In this talk we consider various questions related to a less strongly coupled PDE system for the Mindlin-Timoshenko plate model, viz. the thermoelastic Mindlin-Timoshenko plate model. In this model there is direct coupling between the elastic variables and the thermal variable only in the system of PDEs for the shear angles and in the PDE for the temperature function. Firstly we show that in general uniform stability cannot be attained. This is accomplished by considering a particular configuration in which the plate is rectangular and the PDE system for the plate dynamics is subject to mixed boundary conditions for the shear angle variables and Dirichlet boundary conditions for the displacement and thermal variables. Next we establish the rate of polynomial decay of the model under a condition of radial symmetry when no mechanical dissipation is incorporated in the model. First we consider the case of free boundary conditions for the shear angle variables and Dirichlet boundary conditions for the vertical displacement as well as the thermal variable. It turns out that the rate of decay is slower when Dirichlet boundary conditions are imposed on the elastic variables as well as the thermal variable. Time allowing we will consider the modelling of the heat effects by using Cattaneo’s law and check the effect of this description of the heat effects on the stabilizability of the full model.

Mathematics Subject Classification: 74K10 93B15

Keywords: Mindlin-Timoshenko plate; thermoelastic; polynomial decay

Minisymposion: Mathematical Modeling of Physical Phenomena