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.
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  and Landau and Lifshitz . 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  and Hubert and Schäfer . 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 -convergence techniques Gioia and James , Desimone , Desimone et al. , DeSimone and James , Alicandro and Leone , Kohn and Slastikov , Alouges and Labbé , 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. , Carbou  and Kohn and Slastikov  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 , Valente , Valente and Vergara Caffarelli  and Carbou et al.  where some results concerning global existence of weak solutions are established. Finally, some computational aspects of magnetostriction are presented in Cerimele et al.  and Baňas .
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 , 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 , Ciarlet and Destuynder . Here we try to extend in some sense the work .
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 a bounded open set of . The generic point of is denoted by with . Here and throughout the paper we use bold characters to denote the vector-valued functions. The model combines phenomenological constitutive equations for the magnetization and the displacement . The nonlinear parabolic hyperbolic coupled system describing the dynamics in is given by (see ) where the components of the vector and the tensors are given by Here stand for the components of the linearized strain tensor , and with if and otherwise. The elasticity tensor is assumed to satisfy the following symmetry property, and moreover the inequality holds for some .
As initial and boundary conditions we assume where is the outer unit normal at the boundary .
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 . The unknown , the magnetization vector, is a map from to (the unit sphere of ). The symbol denotes the vector cross product in . Moreover we denote by the components of . The constant represents the damping parameter while is the exchange coefficient. The second equation in (2.1) describes the evolution of the displacement where is a positive constant and is a given external force.
We introduce the functional defined as and put () In the sequel and without loss of generality, we assume that .
Lemma 2.1 (energy estimate). Let with a.e in , then there exist positive constants and depending on such that the following estimate holds:
Proof. We refer the reader to Valente and Vergara Caffarelli . We mention that to obtain the above energy estimate it is assumed that .
Theorem 2.2 (global existence). (Valente and Vergara Caffarelli ). Given and with a.e. in , there exists a weak solution to the problems (2.1)–(2.6) in the sense that (i) with a.e. in ,(ii)for each couple such that vanishing at and , and , one has Moreover the energy estimate (2.9) holds true.
In the sequel, the operators , and will represent divergence, gradient, and laplacian operators, respectively, with respect to the variable .
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 . So the model (2.1) reduces to the following system (we let and ): coupled with in , where the vector is given by The associated initial and boundary conditions writes
4. Dimensional Reduction
4.1. Flat Domains
Let be a real parameter taking values in a sequence of positive numbers converging to zero. We consider flat magnetoelastic domains represented by , where is a regular and bounded subset of . We shall be interested in getting the asymptotic behaviour of the solutions, when .
4.1.1. Scaling and Uniform Bounds
Let be the fields associated with . The scaled equations satisfied by are the following: The vector is defined by
For the scaled displacement we have
The associated energy , defined in (2.7), becomes The energy equation remains unchanged as well as the saturation constraint on magnetization (see (2.5)) which is written as for almost every . The following estimates hold true for all where is given by To get uniform bounds for the solutions we discuss the admissibility criterion for the initial data. An initial datum is said to be admissible if we have The admissibility criterion means Thus, since a.e., to satisfy the criterion, we assume that there exists independent of such that Condition (4.11) means that the couple is essentially independent of the variable and its strong limit is independent of .
Remark 4.1. If the initial data are not admissible, then we expect an initial layer to occur when tends to zero.
4.1.2. Passing to the Limit
Let be a solution of the problem associated to an admissible initial datum . We have Moreover is independent of . For subsequences, the solutions verify the convergences Hence, the couple is independent of the variable . By Aubin’s compactness results, we have Moreover from the Sobolev embedding theorem , the further compactness result follows Recall that with .
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 and be a regular test functions depending on . Multiplying (4.2) by , each Equation (4.4) by and integrating by parts, we get the weak formulations To pass to the limit in these equations we need the following convergence result.
Lemma 4.2. Defining , then where is a function of the variable .
Proof. We multiply the first equation of (4.18) by and choose independent of . We get Hence, passing to the limit, by using convergences (4.14), (4.15), and (4.16), we deduce that the weak- limit of the sequence satisfies which allows to get (4.19).
Remark 4.3. In the sequel and without loss of generality we will assume that .
Now we are able to pass to the limit. We set . We choose in the above weak formulations test functions of the form 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, and . Hence we first get Next, we have We also get Recall that To pass to the limit in the term with 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 Next, we have We also get 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 be a solution of the problem associated with the admissible initial datum . Then, one has strongly in weakly- in and weakly in . The couple is independent of the variable and satisfies in and the following two-dimensional coupled system where The associated initial and boundary conditions are given by where is the weak limit of in .
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 be a solution of the problem associated with the admissible initial datum . Then, one has strongly in weakly- in and weakly in . The couple is independent of the variable and satisfies in and the following one-dimensional coupled system where The associated initial and boundary conditions are given by where is the weak limit of in .
5. Concluding Remarks
We remark that the 1D model obtained in Theorem 4.6 slightly differs from the one derived in . This is simply due to the original 2D model considered in  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 ) where if and otherwise. The parameters , and 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 . 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 with period, for example, of order . So that the three-dimensional domain may be represented by where is a domain of . Various limit models may be obtained depending on the ratio between the size of rugosities and the mean height of the domain.
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