Asymptotic Behavior of Magnetostrictive Elasticity for Microstructured Materials

Abstract

Abstract According to the classical theory of Weiss, Landau, and Lifshitz, in a ferromagnetic body there is a spontaneous magnetization field m, such that ∥m∥ = τ 0 = const in all points of this material Ω. In any stationary configuration, this ferromagnetic body consists of areas (Weiss domains) in which the magnetization is uniform (i.e. m = const) separated by thin transition layers (Bloch walls). Such stationary configuration corresponds to the minimum point of the magnetostrictive free energy E. We are considering an elastic magnetostrictive body in our paper. The elastic magnetostrictive free energy Eδ depends on a small parameter δ such that δ → 0. As usual, the displacement field is denoted by u. We will show that each sequence of minimizers (ui, mi ) contains a subsequence that converges to a couple of fields (u 0, m 0). By means of a Γ-limit procedure we will show that this couple (u 0, m 0) is a minimizer of the new functional E 0. This new functional E 0 describes the magnetic-elastic properties of the body with microstructure.