Thin Film Limits in Magnetoelastic Interactions

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.


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 threedimensional 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

The 3D Model and Preliminary Results
To describe the model equations we consider Ω a bounded open set of R 3 .The generic point of Ω is denoted by x, x 3 with x x 1 , x 2 .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 where the components of the vector l M, u and the tensors S u , L M are given by
As initial and boundary conditions we assume 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 S 2 the unit sphere of R 3 .The symbol × denotes the vector cross product in R 3 .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 and put τ > 0 In the sequel and without loss of generality, we assume that f ≡ 0.
then there exist positive constants c 1 and c 2 depending on T such that the following estimate holds: 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 in Ω, there exists a weak solution M, u to the problems 2.1 -2.6 in the sense that 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.

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 τ 2 0, λ 1 0, τ 1 2τ and λ 2 λ 3 λ : The associated initial and boundary conditions writes

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 R 2 .We shall be interested in getting the asymptotic behaviour of the solutions, when ε → 0.

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 x 1 , x 2 , x 3 x, y, εz with x, y, z X ∈ Ω ω × 0, 1 .For functions R x 1 , x 2 , x 3 and S x 1 , x 2 , x 3 defined in Ω ε we introduce the functions r ε x, y, z and s ε x, y, z defined on Ω by setting Let m ε , w ε be the fields associated with M, W .The scaled equations satisfied by m ε are the following: The vector V ε is defined by For the scaled displacement u ε 0, 0, w ε we have

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The associated energy E ε t , defined in 2.7 , becomes

4.5
The energy equation remains unchanged as well as the saturation constraint on magnetization see 2.5 which is written as for almost every t, X .The following estimates hold true for all t ≥ 0 where E ε 0 is given by

4.8
To get uniform bounds for the solutions we discuss the admissibility criterion for the initial data.An initial datum m ε 0 , w ε 0 is said to be admissible if we have The admissibility criterion means

4.10
Thus, since |m ε 0 | 2 1 a.e., to satisfy the criterion, we assume that there exists C > 0 independent of ε such that

4.11
Condition 4.11 means that the couple m ε 0 , w ε 0 is essentially independent of the variable z and its strong limit m 0 , w 0 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.

Passing to the Limit
Let m ε , w ε be a solution of the problem associated to an admissible initial datum m ε 0 , w ε 0 .We have

4.12
Moreover m 0 x, z m 0 x is independent of z.For subsequences, the solutions verify the convergences

4.14
Hence, the couple m, w is independent of the variable z.By Aubin's compactness results, we have Moreover from the Sobolev embedding theorem W 1,2 Q → L q Q 2 ≤ q ≤ 6 , the further compactness result follows 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

4.18
To pass to the limit in these equations we need the following convergence result.

Lemma 4.2. Defining Θ
where K is a function of the variable x.
Proof.We multiply the first equation of 4.18 by ε and choose

4.20
Mathematical Problems in Engineering 9 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 λm 2 3 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 ψ 0 t, x εψ t, x, εz , g ε t, x, z g t, x εg 0 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, ∂ z ψ ε ε 2 ∂ z ψ εz and ∂ x ψ ε ∂ x ψ 0 ε∂ x ψ εz .Hence we first get

4.22
Next, we have We also get Recall that 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

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Next, we have

4.27
We also get

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 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 ε 0 , w ε 0 .Then, one has m ε , w ε → m, w strongly in w weakly in L 2 R , H 1 0 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

4.32
where w 1 is the weak limit of w ε 1 in L 2 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 4.33

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 ε 0 , w ε 0 .Then, one has m ε , w ε → m, w strongly in L 2 loc R , L 2 Ω , m ε m weakly-in L ∞ R , H 1 Ω and w ε w weakly in L 2 R , H 1 0 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

4.36
where w 1 is the weak limit of w ε 1 in L 2 Ω .

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 where δ ijkl 1 if i j k l and δ ijkl 0 otherwise.The parameters τ 1 , τ 2 , λ 1 , λ 2 , and λ 3 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 ε 2 ε > 0 .So that the three-dimensional domain may be represented by Ω ε { x, z ∈ R 2 × R; x ∈ ω, 0 < z < h ε x } where ω is a domain of R 2 .Various limit models may be obtained depending on the ratio between the size of rugosities and the mean height of the domain.