Thin Film Limits in Magnetoelastic Interactions

Abstract

This paper deals with classical dimensional reductions 3D-2D and 3D-1D in magnetoelastic interactions. We adopt a model described by the Landau-Lifshitz-Gilbert equation for the magnetization field coupled to an evolution equation for the displacement. We identify the limit problem both for flat and slender media by using the so-called energy method.

1. Introduction

Magnetostrictive solids are those in which reversible elastic deformations are caused by changes in the magnetization. These materials have a coupling of ferromagnetic energies with elastic energies. Magnetostriction is observed, to differing degrees, in all ferromagnetic materials. To explain the observed magnetic behaviour, there have been a number of theoretical developments for magnetostrictive materials, including the works by Brown [1] and Landau and Lifshitz [2]. A well-established variational model, called Micromagnetics [1, 2], is in principle available to describe the magnetomechanical response of magnetostrictive solids. Treatments on micromagnetic processes are also available in Aharoni [3] and Hubert and Schäfer [4]. The general micromagnetic problem for reasonably large samples is, however, difficult. That is because of the necessity of resolving exceedingly complex three-dimensional domain structures. For sufficiently small thin films, numerical simulations are now routinely used to explore the energy landscape. But simulations are simply experiments. To interpret them, it is natural to do analysis as well. The understanding of thin film behaviour has been helped by the mathematical asymptotic analysis of energies defined on three-dimensional domains of vanishing thickness, through the use of Γ-convergence techniques Gioia and James [5], Desimone [6], Desimone et al. [7], DeSimone and James [8], Alicandro and Leone [9], Kohn and Slastikov [10], Alouges and Labbé [11], Chipot et al. [12, 13]. The focus in all these papers is energy minimization both with and without magnetostriction, but the dynamics and switching of thin film ferromagnets is also an important topic. The papers by Ammari et al. [14], Carbou [15] and Kohn and Slastikov [16] propose a two-dimensional reduction of the Landau-Lifshitz-Gilbert (LLG) equations of micromagnetic switching. For dynamics including magnetostriction we refer, for example, to Visintin [17], Valente [18], Valente and Vergara Caffarelli [19] and Carbou et al. [20] where some results concerning global existence of weak solutions are established. Finally, some computational aspects of magnetostriction are presented in Cerimele et al. [21] and Baňas [22].

The main goal of this paper is to combine the ideas of the references cited above from the elasticity and micromagnetics literature to derive asymptotic models for magnetostrictive films and nanowires. We are concerned with the passage 3D-2D and 3D-1D in the dynamical theory of thin magnetoelastic films. Our investigation has its starting point in the work by Valente and Vergara Caffarelli [19], where the authors establish the existence of weak solutions to a three-dimensional model for the LLG equation with magnetostriction. We shall assume antiplane displacement. We intend to analyze the behaviour of these solutions with one and two diminishing edges. In order to identify the limit problem we make use of the scaling techniques which are well known in elasticity, see, for example, Ciarlet [23], Ciarlet and Destuynder [24]. Here we try to extend in some sense the work [25].

This paper is organized as follows. In Section 2 we present the general three-dimensional model with corresponding energy estimate and a global existence result of weak solutions. Section 3 describes the antiplane model equations which will be the subject of an asymptotic study. In Section 4, we first consider the dimensional reduction from 3D to 2D. We introduce the natural scaling for the problem and prove uniform bounds for the solutions, with respect to vanishing parameter, which allows us to identify the limit problem. We then consider the 3D-1D reduction and state the limit problem. The last section concludes the paper and provides future directions for this work.

2. The 3D Model and Preliminary Results

To describe the model equations we consider Ω a bounded open set of ℝ³. The generic point of Ω is denoted by (𝑥,𝑥₃) with 𝑥=(𝑥₁,𝑥₂). Here and throughout the paper we use bold characters to denote the vector-valued functions. The model combines phenomenological constitutive equations for the magnetization 𝐌 and the displacement 𝐮. The nonlinear parabolic hyperbolic coupled system describing the dynamics in 𝑄=(0,𝑇)×Ω is given by (see [19]) 𝛾⁻¹𝜕_𝑡𝐌−𝐌×𝑎Δ𝐌−𝜕_𝑡𝐌−𝐥(𝐌,𝐮)=0,𝜌𝜕_𝑡𝑡𝐒(𝐮)+1𝐮−div2𝐋(𝐌)+𝐟=0, (2.1) where the components of the vector 𝐥(𝐌,𝐮) and the tensors 𝐒(𝐮),𝐋(𝐌) are given by 𝑙_𝑖(𝐌,𝐮)=𝜆_{𝑖𝑗𝑘𝑙}𝑀_𝑗𝜀_𝑘𝑙(𝐮),𝐒_𝑘𝑙(𝐮)=𝜎_{𝑖𝑗𝑘𝑙}𝜀_𝑖𝑗(𝐮),𝐋_𝑘𝑙(𝐌)=𝜆_{𝑖𝑗𝑘𝑙}𝑀_𝑖𝑀_𝑗. (2.2) Here 𝜀_𝑖𝑗 stand for the components of the linearized strain tensor 𝜺, 𝜆_{𝑖𝑗𝑘𝑙}=𝜆₁𝛿_{𝑖𝑗𝑘𝑙}+𝜆₂𝛿_𝑖𝑗𝛿_𝑘𝑙+𝜆₃(𝛿_𝑖𝑘𝛿_𝑗𝑙+𝛿_𝑖𝑙𝛿_𝑗𝑘) and 𝜎_{𝑖𝑗𝑘𝑙}=𝜏₁(𝛿_{𝑖𝑗𝑘𝑙}−𝛿_𝑖𝑗𝛿_𝑘𝑙+𝛿_𝑖𝑘𝛿_𝑗𝑙)+𝜏₂𝛿_𝑖𝑗𝛿_𝑘𝑙  with 𝛿_{𝑖𝑗𝑘𝑙}=1 if 𝑖=𝑗=𝑘=𝑙 and 𝛿_{𝑖𝑗𝑘𝑙}=0 otherwise. The elasticity tensor 𝜎_{𝑖𝑗𝑘𝑙} is assumed to satisfy the following symmetry property, 𝜎_{𝑖𝑗𝑘𝑙}=𝜎_{𝑗𝑖𝑘𝑙}=𝜎_{𝑘𝑙𝑖𝑗}, (2.3) and moreover the inequality 𝜎_{𝑘𝑙𝑖𝑗}𝜀_𝑘𝑙𝜀_𝑖𝑗≥𝛽𝜀_𝑘𝑙² (2.4) holds for some 𝛽>0.

As initial and boundary conditions we assume 𝐮(0,⋅)=𝐮₀,𝜕_𝑡𝐮(0,⋅)=𝐮₁,𝐌(0,⋅)=𝐌₀,𝐌₀=1inΩ,𝐮=0,𝜕_𝜈𝐌=0onΣ=(0,𝑇)×𝜕Ω, (2.5)(2.6) where 𝜈 is the outer unit normal at the boundary 𝜕Ω.

The first equation in (2.1), well known in the literature, is the modified LLG equation. The modification lies in the presence of the term 𝐥(𝐌,𝐮). The unknown 𝐌, the magnetization vector, is a map from Ω to 𝒮² (the unit sphere of ℝ³). The symbol × denotes the vector cross product in ℝ³. Moreover we denote by 𝑀_𝑖,𝑖=1,2,3 the components of 𝐌. The constant 𝛾 represents the damping parameter while 𝑎 is the exchange coefficient. The second equation in (2.1) describes the evolution of the displacement 𝐮 where 𝜌 is a positive constant and 𝐟 is a given external force.

We introduce the functional ℰ(𝑡) defined as ℰ(𝑡)=𝑎2_Ω|∇𝐌|²𝛽dΩ+4_Ω|𝜺(𝐮)|²𝜌dΩ+2_Ω𝜕_𝑡𝐮²dΩ (2.7) and put (𝜏>0) ℰ₀=𝑎2_Ω∇𝐌₀²𝜏dΩ+4_Ω∇𝐮₀²𝜌dΩ+2_Ω𝐮₁²dΩ. (2.8) In the sequel and without loss of generality, we assume that 𝐟≡0.

Lemma 2.1 (energy estimate). Let 𝐮₀∈𝐻¹₀(Ω,ℝ³),𝐮₁∈𝐿²(Ω,ℝ³),𝐌₀∈𝐻¹(Ω,ℝ³) with |𝐌₀|=1 a.e in Ω, then there exist positive constants 𝑐₁ and 𝑐₂ depending on 𝑇 such that the following estimate holds: ℰ(𝑡)+^𝑡_0Ω𝜕_𝑡𝐌²dΩd𝑡≤𝑐₁ℰ₀+𝑐₂. (2.9)

Proof. We refer the reader to Valente and Vergara Caffarelli [19]. We mention that to obtain the above energy estimate it is assumed that 𝜎_{𝑖𝑗𝑘𝑙}𝜀_𝑖𝑗(𝐮)𝜀_𝑘𝑙(𝐮)≤𝜏|∇𝐮|².

We have the following global existence result for problem (2.1)–(2.6).

Theorem 2.2 (global existence). (Valente and Vergara Caffarelli [19]). Given 𝐮₀∈𝐻¹(Ω),𝐮₁∈𝐿²(Ω) and 𝐌₀∈𝐻¹(Ω) with |𝐌₀|=1 a.e. in Ω, there exists a weak solution (𝐌,𝐮) to the problems (2.1)–(2.6) in the sense that (i)𝐌∈𝐻¹(𝑄) with |𝐌|=1 a.e. in 𝑄,𝐮∈𝐿²(0,𝑇;𝐻¹₀(Ω)),𝜕_𝑡𝐮∈𝐿²(0,𝑇;𝐿²(Ω)),(ii)for each couple (𝐩,𝐠) such that 𝐩∈𝐶^∞(𝑄) vanishing at 𝑡=0 and 𝑡=𝑇, and 𝐠∈𝐻¹(𝑄)∩𝐶₀(𝑄), one has _𝑄𝛾⁻¹𝜕_𝑡𝐌⋅𝐩+𝑎²_𝑗=1𝐌×𝜕_𝑥𝑗𝐌⋅𝜕_𝑥𝑗𝐩+𝐌×𝜕_𝑡𝐌+𝐥(𝐌,𝐮)⋅𝐩dΩd𝑡=0,_𝑄−𝜌𝜕_𝑡𝐮𝜕_𝑡𝐒(𝐮)+1𝐠+2𝐋(𝐦)∶𝜀(𝐠)dΩd𝑡=0, (2.10)(2.11)Moreover the energy estimate (2.9) holds true.

In the sequel, the operators div,grad, and Δ will represent divergence, gradient, and laplacian operators, respectively, with respect to the variable 𝑥.

3. The Antiplane Case

Our asymptotic analysis will be performed in the framework of a simplified model in which we neglect the in-plane components of displacement; that is, we assume that 𝐮=(0,0,𝑊). So the model (2.1) reduces to the following system (we let 𝜏₂=0,𝜆₁=0,𝜏₁=2𝜏 and 𝜆₂=𝜆₃=𝜆): 𝛾⁻¹𝜕_𝑡𝐌−𝐌×𝑎Δ𝐌−𝜕_𝑡𝐌−𝜆𝐕=0 (3.1) coupled with 𝜌𝜕_𝑡𝑡𝑊−𝜏Δ𝑊−2𝜏𝜕_𝑥3𝑥3𝑊−𝜆𝜕_𝑥1𝑀₁𝑀₃+𝜕_𝑥2𝑀₂𝑀₃+𝜕_𝑥3𝑀²₃=0,2𝜏𝜕_𝑥1𝑥3𝑊−𝜆𝜕_𝑥1𝑀²₁−𝜆𝜕_𝑥2𝑀₁𝑀₂−𝜏𝜕_𝑥3𝜕_𝑥1𝑊−𝜆𝜕_𝑥3𝑀₁𝑀₃=0,−𝜆𝜕_𝑥1𝑀₁𝑀₂+2𝜏𝜕_𝑥2𝑥3𝑊−𝜆𝜕_𝑥2𝑀²₂−𝜏𝜕_𝑥3𝑥2𝑊−𝜆𝜕_𝑥3𝑀₂𝑀₃=0 (3.2) in 𝑄=(0,𝑇)×Ω, where the vector 𝐕=𝐕(𝐌,∇𝑊) is given by 𝐕=𝑀₃𝜕_𝑥1𝑊+𝑀₁𝜕_𝑥3𝑊,𝑀₃𝜕_𝑥2𝑊+𝑀₂𝜕_𝑥3𝑊,𝑀₁𝜕_𝑥1𝑊+𝑀₂𝜕_𝑥2𝑊+3𝑀₃𝜕_𝑥3𝑊. (3.3) The associated initial and boundary conditions writes 𝑊(0,⋅)=𝑊₀,𝜕_𝑡𝑊(0,⋅)=𝑊₁,𝐌(0,⋅)=𝐌₀,𝐌₀=1inΩ,𝑊=0,𝜕_𝜈𝐌=0onΣ=(0,𝑇)×𝜕Ω. (3.4)(3.5)

4. Dimensional Reduction

4.1. Flat Domains

Let 𝜀 be a real parameter taking values in a sequence of positive numbers converging to zero. We consider flat magnetoelastic domains represented by Ω^𝜀=𝜔×(0,𝜀), where 𝜔 is a regular and bounded subset of ℝ². We shall be interested in getting the asymptotic behaviour of the solutions, when 𝜀→0.

4.1.1. Scaling and Uniform Bounds

Let (𝐌,𝑊) be a solution of the problem (3.1)–(3.5) posed in Ω^𝜀. We introduce the change of variables (𝑥₁,𝑥₂,𝑥₃)=(𝑥,𝑦,𝜀𝑧) with (𝑥,𝑦,𝑧)=𝑋∈Ω=𝜔×(0,1). For functions 𝐑(𝑥₁,𝑥₂,𝑥₃) and 𝑆(𝑥₁,𝑥₂,𝑥₃) defined in Ω^𝜀 we introduce the functions 𝐫^𝜀(𝑥,𝑦,𝑧) and 𝑠^𝜀(𝑥,𝑦,𝑧) defined on Ω by setting 𝐑𝑥₁,𝑥₂,𝑥₃=𝐫^𝜀𝑥,𝑦,𝑧;𝑆𝑥₁,𝑥₂,𝑥₃=𝑠^𝜀𝑥,𝑦,𝑧. (4.1)

Let (𝐦^𝜀,𝑤^𝜀) be the fields associated with (𝐌,𝑊). The scaled equations satisfied by (𝐦^𝜀) are the following: 𝛾⁻¹𝜕_𝑡𝐦^𝜀−𝐦^𝜀×𝑎Δ𝐦^𝜀+1𝜀²𝜕_𝑧𝑧𝐦^𝜀−𝜕_𝑡𝐦^𝜀−𝜆𝐕^𝜀=0. (4.2) The vector 𝐕^𝜀 is defined by 𝐕^𝜀=𝑚^𝜀₃𝜕_𝑥𝑤^𝜀+1𝜀𝑚^𝜀₁𝜕_𝑧𝑤^𝜀,𝑚^𝜀₃𝜕_𝑦𝑤^𝜀+1𝜀𝑚^𝜀₂𝜕_𝑧𝑤^𝜀,𝑚^𝜀₁𝜕_𝑥𝑤^𝜀+𝑚^𝜀₂𝜕_𝑦𝑤^𝜀+3𝜀𝑚^𝜀₃𝜕_𝑧𝑤^𝜀. (4.3)

For the scaled displacement 𝐮^𝜀=(0,0,𝑤^𝜀) we have 𝜌𝜕_𝑡𝑡𝑤^𝜀−𝜏Δ𝑤^𝜀+2𝜀²𝜕_𝑧𝑧𝑤^𝜀−𝜆𝜕_𝑥𝑚^𝜀₁𝑚^𝜀₃+𝜕_𝑦𝑚^𝜀₂𝑚^𝜀₃−𝜆𝜀𝜕_𝑧𝑚^𝜀₃²2𝜏=0,𝜀𝜕_𝑥𝑧𝑤^𝜀−𝜆𝜕_𝑥𝑚^𝜀₁²−𝜆𝜕_𝑦𝑚^𝜀₁𝑚^𝜀₂−𝜏𝜀𝜕_𝑧𝑥𝑤^𝜀−𝜆𝜀𝜕_𝑧𝑚^𝜀₁𝑚^𝜀₃=0,−𝜆𝜕_𝑥𝑚^𝜀₁𝑚^𝜀₂𝜏+2𝜀𝜕_𝑦𝑧𝑤^𝜀−𝜆𝜕_𝑦𝑚^𝜀₂²−𝜏𝜀𝜕_𝑧𝑥𝑤^𝜀−𝜆𝜀𝜕_𝑧𝑚^𝜀₂𝑚^𝜀₃=0. (4.4)

The associated energy ℰ^𝜀(𝑡), defined in (2.7), becomes ℰ^𝜀(𝑡)=𝑎2_Ωgrad𝐦^𝜀2𝑎dΩ+2𝜀²_Ω𝜕_𝑧𝐦^𝜀2+𝛽dΩ4_Ωgrad𝑤^𝜀2𝛽dΩ+4𝜀²_Ω𝜕_𝑧𝑤^𝜀2𝜌dΩ+2_Ω𝜕_𝑡𝑤^𝜀2dΩ. (4.5) The energy equation remains unchanged as well as the saturation constraint on magnetization (see (2.5)) which is written as 𝐦^𝜀(𝑡,𝑋)²=𝐦^𝜀₀(𝑋)²=1 (4.6) for almost every (𝑡,𝑋). The following estimates hold true for all 𝑡≥0ℰ^𝜀(𝑡)+^𝑡_0Ω𝜕_𝑡𝐦^𝜀2dΩd𝑡≤𝑐₁ℰ^𝜀₀+𝑐₂, (4.7) where ℰ^𝜀₀ is given by ℰ^𝜀₀=𝑎2_Ωgrad𝐦^𝜀₀²𝑎dΩ+2𝜀²_Ω𝜕_𝑧𝑚^𝜀₀²+𝜏dΩ4_Ωgrad𝑤^𝜀₀²𝜏dΩ+4𝜀²_Ω𝜕_𝑧𝑤^𝜀₀²𝜌dΩ+2_Ω𝑤^𝜀₁²dΩ. (4.8) To get uniform bounds for the solutions we discuss the admissibility criterion for the initial data. An initial datum (𝐦^𝜀₀,𝑤^𝜀₀) is said to be admissible if we have ℰ^𝜀₀<+∞. (4.9) The admissibility criterion means 𝑎2_Ωgrad𝐦^𝜀₀²𝑎dΩ+2𝜀²_Ω𝜕_𝑧𝑚^𝜀₀²+𝜏dΩ4_Ωgrad𝑤^𝜀₀²𝜏dΩ+4𝜀²_Ω𝜕_𝑧𝑤^𝜀₀²𝜌dΩ+2_Ω𝑤^𝜀₁²dΩ<+∞. (4.10) Thus, since |𝐦^𝜀₀|²=1 a.e., to satisfy the criterion, we assume that there exists 𝐶>0 independent of 𝜀 such that grad𝐦^𝜀_0𝐿2(Ω)≤𝐶,𝜕_𝑧𝐦^𝜀_0𝐿2(Ω)≤𝐶𝜀,𝐦^𝜀₀𝑥,𝑦²=1a.e.,grad𝑤^𝜀_0𝐿2(Ω)≤𝐶,𝜕_𝑧𝑤^𝜀_0𝐿2(Ω)≤𝐶𝜀,𝑤^𝜀_1𝐿2(Ω)≤𝐶. (4.11) Condition (4.11) means that the couple (𝐦^𝜀₀,𝑤^𝜀₀) is essentially independent of the variable 𝑧 and its strong limit (𝐦₀,𝑤₀) is independent of 𝑧.

Remark 4.1. If the initial data are not admissible, then we expect an initial layer to occur when 𝜀 tends to zero.

4.1.2. Passing to the Limit

Let (𝐦^𝜀,𝑤^𝜀) be a solution of the problem associated to an admissible initial datum (𝐦^𝜀₀,𝑤^𝜀₀). We have 𝐦^𝜀₀⇀𝐦₀weakly-⋆in𝐿^∞(Ω)andweaklyin𝐻¹(Ω),𝑤^𝜀₀⇀𝑤₀weaklyin𝐻¹(Ω). (4.12) Moreover 𝐦₀(𝑥,𝑧)=𝐦₀(𝑥) is independent of 𝑧. For subsequences, the solutions verify the convergences 𝐦^𝜀⇀𝐦weakly-⋆in𝐿^∞ℝ⁺×Ω∩𝐿^∞ℝ⁺,𝐻¹(Ω,𝑤_𝜀)⇀𝑤weaklyin𝐿²0,𝑇,𝐻¹₀(Ω),𝜕_𝑧𝐦^𝜀⟶0stronglyin𝐿^∞ℝ⁺,𝐿²(Ω),𝜕_𝑧𝑤^𝜀⟶0stronglyin𝐿^∞ℝ⁺,𝐿²(Ω),𝜕_𝑡𝐦^𝜀⇀𝜕_𝑡𝐦weaklyin𝐿²ℝ⁺,𝐿²(Ω),𝜕_𝑡𝑤^𝜀⇀𝜕_𝑡𝑤weaklyin𝐿²0,𝑇;𝐿²(Ω). (4.13)(4.14) Hence, the couple (𝐦,𝑤) is independent of the variable 𝑧. By Aubin’s compactness results, we have 𝐦^𝜀,𝑤^𝜀⟶(𝐦,𝑤)stronglyin𝐿²_locℝ⁺,𝐿²(Ω). (4.15) Moreover from the Sobolev embedding theorem 𝑊^1,2(𝑄)→𝐿^𝑞(𝑄)(2≤𝑞≤6), the further compactness result follows 𝑚^𝜀_𝑖𝑚^𝜀_𝑗⟶𝑚_𝑖𝑚_𝑗stronglyin𝐿²(𝑄),𝑖,𝑗=1,2,3. (4.16) Recall that 𝑄=(0,𝑇)×Ω with Ω=𝜔×(0,1).

In order to pass to the limit we look at the variational formulation of the scaled problem (4.2)–(4.4) by using an oscillating test functions. Let 𝜓^𝜀(𝑡,𝑥,𝑧) and 𝑔^𝜀(𝑡,𝑥,𝑧) be a regular test functions depending on 𝜀. Multiplying (4.2) by 𝜓^𝜀, each Equation (4.4) by 𝑔^𝜀 and integrating by parts, we get the weak formulations 𝛾⁻¹_𝑄𝜕_𝑡𝐦^𝜀⋅𝜓^𝜀dΩd𝑡+_𝑄𝐦^𝜀×𝜕_𝑡𝐦^𝜀⋅𝜓^𝜀dΩd𝑡=−𝜆_𝑄𝐦^𝜀×𝐕^𝜀⋅𝜓^𝜀dΩd𝑡−𝑎_𝑄𝐦^𝜀×grad𝐦^𝜀⋅grad𝜓^𝜀−𝑎dΩd𝑡𝜀²_𝑄𝐦^𝜀×𝜕_𝑧𝐦^𝜀⋅𝜕_𝑧𝜓^𝜀dΩd𝑡,−𝜌_𝑄𝜕_𝑡𝑤^𝜀𝜕_𝑡𝑔^𝜀dΩd𝑡+𝜏_𝑄grad𝑤^𝜀grad𝑔^𝜀dΩd𝑡+2𝜏𝜀²_𝑄𝜕_𝑧𝑤^𝜀𝜕_𝑧𝑔^𝜀dΩd𝑡+𝜆_𝑄𝑚^𝜀₁𝑚^𝜀₃𝜕_𝑥𝑔^𝜀dΩd𝑡+𝜆_𝑄𝑚^𝜀₂𝑚^𝜀₃𝜕_𝑦𝑔^𝜀𝜆dΩd𝑡+𝜀_𝑄𝑚^𝜀₃²𝜕_𝑧𝑔^𝜀𝜏dΩd𝑡=0,−2𝜀_𝑄𝜕_𝑧𝑤^𝜀𝜕_𝑥𝑔^𝜀dΩd𝑡+𝜆_𝑄𝑚^𝜀₁²𝜕_𝑥𝑔^𝜀dΩd𝑡+𝜆_𝑄𝑚^𝜀₁𝑚^𝜀₂𝜕_𝑦𝑔^𝜀+𝜏dΩd𝑡𝜀_𝑄𝜕_𝑥𝑤^𝜀𝜕_𝑧𝑔^𝜀𝜆dΩd𝑡+𝜀_𝑄𝑚^𝜀₁𝑚^𝜀₂𝜕_𝑧𝑔^𝜀𝜆dΩd𝑡=0,_𝑄𝑚^𝜀₁𝑚^𝜀₂𝜕_𝑥𝑔^𝜀dΩd𝑡−2𝜏𝜀_𝑄𝜕_𝑧𝑤^𝜀𝜕_𝑦𝑔^𝜀dΩd𝑡+𝜆_𝑄𝑚^𝜀₂²𝜕_𝑦𝑔^𝜀+𝜏dΩd𝑡𝜀_𝑄𝜕_𝑥𝑤^𝜀𝜕_𝑧𝑔^𝜀𝜆dΩd𝑡+𝜀_𝑄𝑚^𝜀₂𝑚^𝜀₃𝜕_𝑧𝑔^𝜀dΩd𝑡=0. (4.17)(4.18) To pass to the limit in these equations we need the following convergence result.

Lemma 4.2. Defining Θ^𝜀∶=(1/𝜀)𝜕_𝑧𝑤^𝜀, then Θ^𝜀𝜆⇀Θ=−𝑚2𝜏³²+𝐾weakly-⋆in𝐿^∞ℝ⁺,𝐿²(Ω), (4.19) where 𝐾 is a function of the variable 𝑥.

Proof. We multiply the first equation of (4.18) by 𝜀 and choose 𝑔^𝜀=𝑔∈𝒟(𝑄) independent of 𝜀. We get 𝜀−𝜌_𝑄𝜕_𝑡𝑤^𝜀𝜕_𝑡𝑔dΩd𝑡+𝜏_𝑄grad𝑤^𝜀grad𝑔dΩd𝑡+𝜆_𝑄𝑚^𝜀₁𝑚^𝜀₃𝜕_𝑥𝑔dΩd𝑡+𝜆_𝑄𝑚^𝜀₂𝑚^𝜀₃𝜕_𝑦𝑔dΩd𝑡+2𝜏𝜀_𝑄𝜕_𝑧𝑤^𝜀𝜕_𝑧𝑔dΩd𝑡+𝜆_𝑄𝑚^𝜀₃²𝜕_𝑧𝑔dΩd𝑡=0. (4.20) Hence, passing to the limit, by using convergences (4.14), (4.15), and (4.16), we deduce that the weak-⋆ limit Θ of the sequence Θ^𝜀 satisfies 𝜕_𝑧(2𝜏Θ+𝜆𝑚²₃)=0 which allows to get (4.19).

Remark 4.3. In the sequel and without loss of generality we will assume that 𝐾≡0.

Now we are able to pass to the limit. We set 𝑄_𝒯=ℝ⁺×𝜔. We choose in the above weak formulations test functions of the form 𝜓^𝜀(𝑡,𝑥,𝑧)=𝜓₀()(),𝑔𝑡,𝑥+𝜀𝜓𝑡,𝑥,𝜀𝑧_𝜀(𝑡,𝑥,𝑧)=𝑔(𝑡,𝑥)+𝜀𝑔₀(𝑡,𝑥)ℎ(𝜀𝑧). (4.21) We pass to the limit in each term of (4.17) by using the convergence results (4.14), (4.15), (4.16) and the following facts, 𝜕_𝑧𝜓^𝜀=𝜀²(𝜕_𝑧𝜓)(𝜀𝑧) and 𝜕𝑥𝜓^𝜀=𝜕𝑥𝜓₀+𝜀𝜕𝑥𝜓(𝜀𝑧). Hence we first get _𝑄𝜕_𝑡𝐦^𝜀⋅𝜓^𝜀dΩd𝑡⟶_𝑄𝒯𝜕_𝑡𝐦⋅𝜓₀d𝑥d𝑡,_𝑄𝐦^𝜀×𝜕_𝑡𝐦^𝜀⋅𝜓^𝜀dΩd𝑡⟶_𝑄𝒯𝐦×𝜕_𝑡𝐦⋅𝜓₀d𝑥d𝑡. (4.22) Next, we have _𝑄𝐦^𝜀×grad𝐦^𝜀⋅grad𝜓^𝜀dΩd𝑡⟶_𝑄𝒯𝐦×grad𝐦⋅grad𝜓₀d𝑥d𝑡. (4.23) We also get 1𝜀²_𝑄𝑎𝐦^𝜀×𝜕_𝑧𝐦^𝜀⋅𝜕_𝑧𝜓^𝜀dΩd𝑡⟶0. (4.24) Recall that 𝐕^𝜀=𝑚^𝜀₃𝜕_𝑥𝑤^𝜀+1𝜀𝑚^𝜀₁𝜕_𝑧𝑤^𝜀,𝑚^𝜀₃𝜕_𝑦𝑤^𝜀+1𝜀𝑚^𝜀₂𝜕_𝑧𝑤^𝜀,𝑚^𝜀₁𝜕_𝑥𝑤^𝜀+𝑚^𝜀₂𝜕_𝑦𝑤^𝜀+3𝜀𝑚^𝜀₃𝜕_𝑧𝑤^𝜀. (4.25) To pass to the limit in the term with 𝐕^𝜀 we make use of the convergence of Lemma 4.2.

Similarly we pass to the limit in the weak formulation (4.18). The convergences (4.14) and (4.15) allow to get _𝑄𝜕_𝑡𝑤^𝜀𝜕_𝑡𝑔^𝜀dΩd𝑡⟶_𝑄𝒯𝜕_𝑡𝑤𝜕_𝑡𝑔d𝑥d𝑡,_𝑄grad𝑤^𝜀grad𝑔^𝜀dΩd𝑡⟶_𝑄𝒯grad𝑤grad𝑔d𝑥d𝑡. (4.26) Next, we have _𝑄𝑚^𝜀₁𝑚^𝜀₃𝜕_𝑥𝑔^𝜀dΩd𝑡⟶_𝑄𝒯𝑚₁𝑚₃𝜕_𝑥𝑔d𝑥d𝑡,_𝑄𝑚^𝜀₂𝑚^𝜀₃𝜕_𝑦𝑔^𝜀dΩd𝑡⟶_𝑄𝒯𝑚₂𝑚₃𝜕_𝑦𝑔d𝑥d𝑡. (4.27) We also get 1𝜀²_𝑄𝜕_𝑧𝑤^𝜀𝜕_𝑧𝑔^𝜀1dΩd𝑡⟶0,𝜀_𝑄𝑚^𝜀₃²𝜕_𝑧𝑔^𝜀dΩd𝑡⟶0. (4.28) It remains to identify the limit of the two last equations of (4.18). Let us pass to the limit in the second equation of (4.18). We make use of the Lemma 4.2 to get 1𝜀_𝑄𝜕_𝑧𝑤^𝜀𝜕_𝑥𝑔^𝜀dΩd𝑡⟶−_𝑄𝒯𝜆𝑚2𝜏³²𝜕_𝑥𝑔d𝑥d𝑡. (4.29) Similarly, by the same arguments above, we also get the limit both for the other terms and for the last equation of (4.18).

We proved the result.

Theorem 4.4. Let (𝐦^𝜀,𝑤^𝜀) be a solution of the problem associated with the admissible initial datum (𝐦^𝜀₀,𝑤^𝜀₀). Then, one has (𝐦^𝜀,𝑤^𝜀)→(𝐦,𝑤) strongly in 𝐿²_loc(ℝ⁺,𝐿²(Ω)),𝐦^𝜀⇀𝐦 weakly-⋆ in 𝐿^∞(ℝ⁺,𝐻¹(Ω)) and 𝑤^𝜀⇀𝑤 weakly in 𝐿²(ℝ⁺,𝐻¹₀(Ω)). The couple (𝐦,𝑤) is independent of the variable 𝑧 and satisfies in ℝ⁺×𝜔,|𝐦(𝑡,𝑥)|²=1 and the following two-dimensional coupled system 𝛾⁻¹𝜕_𝑡𝐦−𝐦×𝑎Δ𝐦−𝜕_𝑡𝐦−𝜆𝐕=0,𝜌𝜕_𝑡𝑡𝑤−𝜏Δ𝑤−𝜆𝜕_𝑥𝑚₁𝑚₃−𝜆𝜕_𝑦𝑚₂𝑚₃𝜕=0,_𝑥𝑚²₁+𝑚²₃+𝜕_𝑦𝑚₁𝑚₂𝜕=0,_𝑦𝑚²₂+𝑚²₃+𝜕_𝑥𝑚₁𝑚₂=0, (4.30) where 𝐕=𝑚₃𝜕_𝑥𝜆𝑤−𝑚2𝜏¹𝑚²₃,𝑚₃𝜕_𝑦𝜆𝑤−𝑚2𝜏²𝑚²₃,𝑚₁𝜕_𝑥𝑤+𝑚₂𝜕_𝑦𝑤−3𝜆𝑚2𝜏³³. (4.31) The associated initial and boundary conditions are given by 𝑤(0,𝑥)=𝑤₀,𝜕_𝑡𝑤(0,𝑥)=𝑤₁,𝐦(0,𝑥)=𝐦₀,𝐦₀=1in𝜔𝑤=0,𝜕_𝜈𝐦=0on𝜕𝜔, (4.32) where 𝑤₁ is the weak limit of 𝑤^𝜀₁ in 𝐿²(Ω).

Remark 4.5. Note that if the function 𝐾 introduced in Lemma 4.2 is such that 𝐾≢0, then the vector 𝐕 in (4.30) becomes 𝐕=𝑚₃𝜕_𝑥𝜆𝑤−𝑚2𝜏²³+𝐾𝑚₁,𝑚₃𝜕_𝑦𝜆𝑤−𝑚2𝜏²³+𝐾𝑚₂,𝑚₁𝜕_𝑥𝑤+𝑚₂𝜕_𝑦𝜆𝑤−3𝑚2𝜏²³+𝐾𝑚₃. (4.33)

4.2. Slender Domains

Note that we can proceed as above to get result for the 3D-1D dimensional reduction. In fact, we have the following theorem which we state without proof.

Theorem 4.6. Let (𝐦^𝜀,𝑤^𝜀) be a solution of the problem associated with the admissible initial datum (𝐦^𝜀₀,𝑤^𝜀₀). Then, one has (𝐦^𝜀,𝑤^𝜀)→(𝐦,𝑤) strongly in 𝐿²_loc(ℝ⁺,𝐿²(Ω)),𝐦^𝜀⇀𝐦 weakly-⋆ in 𝐿^∞(ℝ⁺,𝐻¹(Ω)) and 𝑤^𝜀⇀𝑤 weakly in 𝐿²(ℝ⁺,𝐻¹₀(Ω)). The couple (𝐦,𝑤) is independent of the variable 𝑥 and satisfies in ℝ⁺×(0,1),|𝐦(𝑡,𝑧)|²=1 and the following one-dimensional coupled system 𝛾⁻¹𝜕_𝑡𝐦−𝐦×𝑎𝜕_𝑧𝑧𝐦−𝜕_𝑡𝐦−𝜆𝐕=0,𝜌𝜕_𝑡𝑡𝑤−2𝜏𝜕_𝑧𝑧𝑤−𝜆𝜕_𝑧𝑚²₃=0, (4.34) where −𝜆𝐕=𝜏𝑚₁𝑚²₃+𝑚₁𝜕_𝑧𝜆𝑤,−𝜏𝑚₂𝑚²₃−𝑚₂𝜕_𝑧𝜆𝑤,−𝜏𝑚₁𝑚²₃−𝜆𝜏𝑚₂𝑚²₃+3𝑚₃𝜕_𝑧𝑤. (4.35) The associated initial and boundary conditions are given by 𝑤(0,𝑧)=𝑤₀,𝜕_𝑡𝑤(0,𝑧)=𝑤₁,𝐦(0,𝑧)=𝐦₀,𝐦₀()𝑤=1in0,1𝑡,𝑗=0,𝜕_𝑧𝐦𝑡,𝑗=0for𝑗=0,1, (4.36) where 𝑤₁ is the weak limit of 𝑤^𝜀₁ in 𝐿²(Ω).

5. Concluding Remarks

We remark that the 1D model obtained in Theorem 4.6 slightly differs from the one derived in [25]. This is simply due to the original 2D model considered in [18] which is obtained by making some approximations on the total energy. The limiting behaviours obtained in this work concern magnetoelastic interactions system when we neglect in-plane components of displacement. The present paper is our first attempt at scrutinizing the reduced theories from the complete model. The analysis with arbitrary displacement seems to be more difficult and needs much more investigations. In fact, it would be interesting to consider the general model which consists of the three-dimensional case with total energy (see [18]) ℰ(𝑡)=12_Ω𝑎|∇𝐌|²+𝜏₁|∇𝑈|²+𝜏₂(div𝑈)²+𝜆₁𝛿_{𝑘𝑙𝑖𝑗}∇_𝑖𝑈_𝑗𝑀_𝑘𝑀_𝑙+𝜆₂|𝐌|²div𝑈+2𝜆₃∇𝑈_𝑖⋅𝐌𝑀_𝑖, (5.1) where 𝛿_{𝑖𝑗𝑘𝑙}=1 if 𝑖=𝑗=𝑘=𝑙 and 𝛿_{𝑖𝑗𝑘𝑙}=0 otherwise. The parameters 𝜏₁,𝜏₂,𝜆₁,𝜆₂, and 𝜆₃ are positive constants.

Another direction for future research concerns magnetic domain walls (DWs) which are boundaries in magnetic materials that divide regions with distinct magnetization directions. The manipulation and control of DWs in ferromagnetic nanowires (essentially one dimensional models) has recently become a subject of intense experimental and theoretical research, see, for example, Carbou and Labbé [26]. The rapidly growing interest in the physics of the DW motion can be mainly explained by a promising possibility of using DWs as the basis for next-generation memory and logic devices. However, in order to realize such devices in practice it is essential to be able to position individual DWs precisely along magnetic nanowires. It would be interesting to address within the context of the present paper the stability of the propagation of such processing DWs with respect to perturbations of the initial magnetization profile, some anisotropy properties of the nanowire, and applied magnetic field.

Finally, a valuable direction for future research is the effect of very small domain irregularities on the limiting problems. More precisely, the roughness may be defined by means of a periodical function ℎ^𝜀 with period, for example, of order 𝜀²(𝜀>0). So that the three-dimensional domain may be represented by Ω^𝜀={(𝑥,𝑧)∈ℝ²×ℝ;𝑥∈𝜔,0<𝑧<ℎ^𝜀(𝑥)} where 𝜔 is a domain of ℝ². Various limit models may be obtained depending on the ratio between the size of rugosities and the mean height of the domain.