Effective theories for incompressible magnetoelastic shallow shells

Abstract

This thesis is dedicated to rigorously deriving a reduced two-dimensional model for incompressible magnetoelastic shallow shells. To be precise, we characterize the asymptotic behaviour, in the sense of Gamma-convergence, of a thin magnetoelastic shallow shell as the thickness of the shell goes to zero. This is achieved by following the customary approach for Gamma-convergence. We first show compactness of states with equibounded energies, identify a uniform lower bound for the energies and finally prove the existence of a recovery sequence, thus showing the optimality of the lower bound.The energy of the magnetoelastic shallow shell is the sum of elastic energy, exchange energy and magnetostatic energy and depends on the elastic deformation and magnetization of the material. The elastic energy is defined on the original configuration of the material, whereas the exchange energy and magnetostatic energy are defined on the deformed configuration. As such, the problem presents a mixed Euler-Lagrangian variational structure, and calls for novel techniques. The compactness is achieved up to rigid motions. For deformations, it relies on an approximation by rigid movements, whereasfor magnetizations it is based on a careful consideration of the geometry of the deformed domain. The proof of the lower bound is performed seperately for elastic and exchange energy using the established compactness. Convergence of the magnetostatic energy is shown using an existence result for minimizers and establishing a suitable compactness.Then, the existence of a smooth recovery sequence is constructively shown and combined with a density argument. The result is obtained by a combination of variational methods(Gamma-convergence) with degree theory, fixed-point and geometrical arguments.The proof strategy relies on an adaptation of an analogous result for incompressible magnetoelastic plates from M. Bresciani [1] and an application of some results by I.Velcic on elastic shallow shells [2].