Asymptotic behavior of magnetostrictive elasticity for microstructured materials

• Autorzy: Bojarski J. L. , Wierzbicki E.
• Języki publikacji: EN

Abstrakty

EN

According to the classical theory of Weiss, Landau, and Lifshitz, in a ferromagnetic body there is a spontaneous magnetization field m, such that ∥m∥ = τ⁰ = 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 E_δ 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 (u_i,m_i) contains a subsequence that converges to a couple of fields (u₀,m₀). By means of a Γ-limit procedure we will show that this couple (u₀,m₀) is a minimizer of the new functional E₀. This new functional E₀ describes the magnetic-elastic properties of the body with microstructure.

Słowa kluczowe

EN

homogenization; Gamma-limit procedure; existence of a minimum for free energy; elasticity with microstructure

PL

homogenizacja

• Wydawca: Wydawnictwo Politechniki Łódzkiej
• Czasopismo: Mechanics and Mechanical Engineering
• Rocznik: 2018
• Tom: Vol. 22, nr 3
• Strony: 751--759
• Opis fizyczny: Bibliogr. 8 poz.

Twórcy

autor

Bojarski J. L.

• Faculty of Civil and Environmental Engineering, Warsaw University of Life Sciences, Nowoursynowska 159, Warsaw, Poland

autor

Wierzbicki E.

• Faculty of Civil and Environmental Engineering, Warsaw University of Life Sciences, Nowoursynowska 159, Warsaw, Poland

Bibliografia

• [1] Shu, Y.C., Lin, M.P. andWu, K.C.: Micromagnetic modeling of magnetostrictive materials under intrinsic stress, Mechanics of materials, 36, 10, 975-997, 2004.
• [2] Anzellotti, G., Baldo, S. and Visintin, A.: Asymptotic behavior of the Landau-Lifshitz model of ferromagnetism, Applied Mathematics and Optimization, 23, 1, 171-192, 1991.
• [3] Alber, H.D.: A sharp interface model for phase transformations in crystals-instability caused by microstructure-simulations, Open seminar on partial differentia equations. Warsaw University of Technology, Faculty of Mathematics and Information Sciences, May 17, 2018.
• [4] Giusti, E.: Minimal Surfaces and Functions of Bounded Variation, Birkhäuser, Boston, 1984.
• [5] Leoni, G.: A First Course in Sobolev Spaces. Graduate Studies in Mathematics, vol. 105, American Mathematical Society, Providence, Rhode Island, 2009.
• [6] Adams, R.: Sobolev Spaces, Academic Press, New York, San Francisco, London, 1975.
• [7] Rockafellar, R.T.: Integral functionals, normal integrands and measurable selections, in: Nonlinear Operators and the Calculus of Variations. Lect. Notes Math., 543, Springer, Berlin 1975, 157-207.
• [8] Ekeland, I., Temam, R.: Convex Analysis and Variational Problems, North-Holland, Amsterdam and New York, 1976.

Uwagi

Opracowanie rekordu w ramach umowy 509/P-DUN/2018 ze środków MNiSW przeznaczonych na działalność upowszechniającą naukę (2018).