Magnetoelastic thin films at large strains

Magnetoelasticity describes the mechanical behavior of solids under magnetic effects. The magnetoelastic coupling is based on the presence of small magnetic domains in the material [13]. In the absence of an external magnetic field, these magnetic domains are randomly oriented. When an external magnetic field is applied, the mesostructure of magnetic domains changes by magnetic-domain wall motion, by magnetization-vector rotation, and, for some specific alloys, by magnetic-field-driven martensitic-variant transformation. The net effect is a magnetically induced deformation in the body. Conversely, mechanical deformations modify the magnetic response of a specimen by influencing the magnetic anisotropy of the domains, so that the magnetic and the mechanical behavior of the material are fully coupled. We refer to, e.g., [4, 7, 8, 14] for an exposition on the foundations of magnetoelasticity and to [17] for some related mathematical considerations.

The mathematical modeling of magnetoelasticity is a lively area of research, triggered by the interest in the so-called multifunctional materials. Among these, one has to mention rare-earth alloys such as TerFeNOL and GalFeNOL, as well as ferromagnetic shape-memory alloys as \(\hbox {Ni}_2\)MnGa, NiMnInCo, NiFeGaCo, FePt, FePd, among others [16]. These materials exhibit a remarkable magnetostrictive behavior, for reversible strains as large as 10% can be activated by the imposition of relatively moderate magnetic fields. This strong magnetoelastic coupling makes them relevant in a wealth of innovative applications including sensors and actuators [2].

The aim of this paper is to present a model of a thin film undergoing large strain deformations in the membrane regime. This will be inferred from a variational dimension-reduction procedure from a corresponding three-dimensional model at large strains.

Dimension-reduction techniques play an important role in nonlinear analysis and numeric s, for they allow simpler computational approaches, still preserving the main features of the corresponding bulk model. The last decades have witnessed remarkable progresses on dimension reduction by variational methods, particularly by \(\varGamma \)-convergence [3, 6], together with quantitative rigidity estimates [9]. Among the many results on the elastic response of low-dimensional objects, we mention the rigorous justification of membrane theory [22, 23], bending theory [9, 27], and von Kármán theory [10, 21] for plates as variational limits of nonlinear three-dimensional elasticity for vanishing thickness. In particular, we refer to [10] for the derivation of a hierarchy of different plate models and for a thorough literature review.

A rigorous derivation of a model for magnetic thin films has been first obtained in [11]. A rate-independent evolution of Kirchhoff–Love magnetic plates together with the passage from three-dimensional linearized magnetoelasticity to the corresponding two-dimensional theory is the subject of [19]. Magnetostriction in thin films has been considered, also from the numerical viewpoint, in [24‐26]. With respect to these results, this paper presents a fundamental novelty as it represents the first rigorous analytical treatment including also the large-strain magnetoelastic regime.

In the classical dimension reduction for small-strain elastic thin plates, the analysis is set in cylindrical domains whose heights depend on a thickness parameter eventually tending to zero. The same setting applies in magnetoelasticity. Under the small-deformations assumption, the magnetization may be assumed to be directly defined on the reference configuration. This simplification is, however, not amenable in the large-strain regime, for the magnetization is defined on the deformed configuration instead. The latter is, however, a priori not known, as it depends on the deformation itself. In particular, this naturally leads to a mixed Eulerian–Lagrangian formulation of the problem. Compared with previous small-strain contributions, the mathematical framework of this work is hence much more involved. A distinctive difficulty arises from the need of ensuring that admissible deformations are globally injective. In the bulk, this can be achieved by imposing the so-called Ciarlet–Nečas condition [5]. For films, however, no comparable condition, i.e., allowing for a variational approach, seems to be available. A further difficulty is represented by the Maxwell system, which is formulated in actual space. In order to identify the asymptotic behavior of the stray field, we have to characterize the limiting differential constraints in weak form by keeping track of the deformed configuration.

The main result of the paper is the derivation of a variational model for thin-film specimens as a \(\varGamma \)-limit of a suitably scaled energies of a bulk model for vanishing thickness. In Theorem 3.2, we prove in full generality the \(\varGamma \)–\(\liminf \) inequality, showing that our limit energy functional always represents a lower bound for the asymptotic behavior of the three-dimensional energy functionals. If the limit film deformation is approximately injective in the sense of Definition 3.3, we show that the \(\varGamma \)–\(\liminf \) is indeed the largest lower semicontinuous lower bound for the magnetoelastic-plate functionals as the thickness goes to zero, i.e., it is the \(\varGamma \)-limit; cf.  Theorem 3.4. Here, the approximate injectivity means that there is a sequence of deformations of the bulk which are globally injective and converge in a suitable sense to the film deformation. Additionally, in Theorem 3.5 we prove a complete \(\varGamma \)-convergence result under the additional assumption that the admissible three-dimensional deformations satisfy a suitable injectivity requirement which guarantees that the limit deformation of the film is globally injective.

The paper is organized as follows. In Sect. 2, we introduce the mathematical setting of the problem. Section 3 is devoted to the statements of all results, and Sect. 4 contains all proofs.

We use the standard notation for Sobolev and Lebesgue spaces, i.e., \(W^{k,p} \) and \(L^p \) [1]. If \(A\in {\mathbb {R}}^{3\times 2}\) and \(b\in {\mathbb {R}}^3\) we write \((A|b)\in {\mathbb {R}}^{3\times 3}\) for a matrix whose first two columns are created by the first two columns of A and the third one by the vector b. The set of proper rotations is denoted by SO\((3):=\{R\in {\mathbb {R}}^{3\times 3}:\, R^\top R= RR^\top = \text {Id}, \ \mathrm{det}\, R=1\}\) where Id is the identity matrix.

Let \(\omega \subset {\mathbb {R}}^2\) be a bounded Lipschitz domain representing the planar reference configuration of the film, define the reference configuration of a thin magnetoelastic plate asand set \(\varOmega :=\varOmega _1\). In the expression above, \(h>0\) represents the thickness of the plate, eventually bound to go to zero. Correspondingly, we will consider limits as \(h \rightarrow 0\) of sequences of functionals by means of \(\varGamma \)-convergence [6]. This is a standard approach to characterize the limiting behavior of a sequence of bulk energies for specimens of very small thickness.

Assume that X is a subset of a reflexive Banach space. We say that \(\{I_h\}_{h>0}\) for \(I_h:X\rightarrow {\mathbb {R}}\cup \{\infty \}\) \(\varGamma \)-converges to \(I:X\rightarrow {\mathbb {R}}\cup \{+\infty \}\) if the following conditions hold simultaneously: where the symbol \({\mathop {\rightarrow }\limits ^{X}} \) indicates the convergence with respect to a properly chosen (weak) topology in X. If (2.1) holds, we say that I is the \(\varGamma \)-limit of \(\{I_h\}_{h>0}\) (with respect to that topology).

The state of the magnetoelastic material is defined in terms of its deformation w and its magnetization m. The deformation \(w: \varOmega _h \rightarrow {\mathbb {R}}^3\) is required to belong to \(W^{1,p}(\varOmega _h;{\mathbb {R}}^3)\) for some givento be orientation-preserving, namely, \(\det \nabla w >0\) almost everywhere, and to satisfy the Ciarlet–Nečas condition [5]where \({\mathcal {L}}^3\) stands for the three-dimensional Lebesgue measure. In particular, w is identified with the unique continuous representative in the equivalence class. The magnetization m is set on the open deformed configuration, namely, \(m:\varOmega ^w_h \rightarrow {{\mathbb {S}}}^2\), where \(\varOmega ^w_h\) is given bywhich is well-defined, for w is continuous. The magnetization m is hence required to fulfill the saturation constraint \(|m|=1\) on \(\varOmega _h^w\).

In what follows, for every \(x\in {\mathbb {R}}^3\) in the referential space we write \(x=(x',x_3)\) where \(x'\in {\mathbb {R}}^2\) is referred to as the planar coordinates of x, and we denote by \(\nabla '\) the gradient with respect to such planar coordinates. We use instead the symbol \(\xi \in {\mathbb {R}}^3\) to indicate variable s in the actual space.

Following the approach in [15, 20, 28], we consider the total energy \(I_h\) defined asIn the formula above, \(W:\mathbb {M}^{3\times 3}\times \mathbb {S}^2\rightarrow [0,+\infty )\) is the elastic energy density associated with the plate, which is a continuous function satisfying the following assumptions:for every \(F\in \mathbb {M}^{3\times 3}\), \(R\in \mathrm{SO}(3)\), and \(\lambda \in \mathbb {S}^2\). In fact, assumptions (2.5)–(2.6) are not strictly needed for the analysis, but rather required by modeling considerations.

The second term in the expression of \(I_h\) in (2.3) is the exchange energy. The constant \(\alpha \) is related to the size of ferromagnetic texture. The material is assumed to be of nonsimple type [18]. This is expressed by the occurrence of the third term in \(I_h\), providing a higher-order contribution and a further length scale to the problem. Regarding the fourth term, we will require that \(\varPhi :\mathbb {M}^{3\times 3}\rightarrow [0,+\infty )\) is a continuous map satisfying the following assumptionswhere \(q>\frac{3p}{p-3}\). This last quantification is introduced in [12] and ensures that, for all \(\lambda >0\) and \(w\in W^{2,p}(\varOmega _h;{\mathbb {R}}^3)\) such thatthere exists \(c>0\) depending on \(\lambda >0\) with the property thatNote that the left-hand side of inequality (2.8) is a part of the energy functional (2.3). The last term in (2.3) represents the magnetostatic energy. In particular, \(\mu _0\) is the permittivity of void, and \(u_m\) solves the Maxwell equationwhere \(\chi _{\varOmega _h^w}\) is the characteristic function of the set \(\varOmega _h^w\). For simplicity, we assume that the deformations w satisfy the boundary conditionsTo consider alternative boundary conditions would call for solving some additional technicalities which, we believe, would excessively complicate the argument. We hence leave this extension to some possible further investigation.

This section is devoted to the statement of our main \(\varGamma \)-convergence results. All proofs are postponed to the following Sect. 4.

For notational convenience, for every open set \(U\subset {\mathbb {R}}^2\) we denote by \(\mathring{W}^{k,p}(U;{\mathbb {R}}^n)\) the set of \(W^{k,p}\)-maps having zero mean on U, i.e., \(y\in \mathring{W}^{k,p}(U;{\mathbb {R}}^n)\) if \(y\in W^{k,p}(U;{\mathbb {R}}^n)\) and \(\int _U y(x')\,\mathrm{d}x'=0\). As it is standard, we write \(\mathring{W}^{k,p}(U)\) if \(n=1\).

We first introduce the set \({\mathcal {A}}\) of admissible limiting deformations \(y:\omega \rightarrow {\mathbb {R}}^3\), Cosserat vectors \(b:\omega \rightarrow {\mathbb {R}}^3\), and magnetizations \({\mathscr {M}}: \omega \rightarrow {\mathbb {S}}^2\), defined asLet us first state the following lemma, which will be instrumental in characterizing the limiting stray fields and formulating the limiting functional. As mentioned, the lemma is proved in Sect. 4 below.

The limiting energy is given by the functional \( {\mathcal {F}} :{\mathcal {A}}\rightarrow [0,+\infty )\) defined asfor every \((y,b,{\mathscr {M}})\in {\mathcal {A}}\), where the pair \(\Big ({\mathscr {U}}_{y,b,{\mathscr {M}}}|{\mathscr {V}}_{y,b,{\mathscr {M}}}\Big )\in \mathring{W}^{1,2}(\omega )\times L^2(\omega )\) is the restriction to \(\omega \) of the unique solution to (3.3)–(3.4) in the sense of Lemma 3.1.

We start by providing a lower bound for the asymptotic behavior of the functionals \(\{E_h\}_h\) along sequences of deformations and magnetizations with equibounded energies. Again, the proof is postponed to Sect. 4.

The statement of our second main result requires the specification of the class of admissible deformations. This is given through the following definition.

The deformations in Fig. 1 and on the right of Fig. 2 fulfill the requirements of Definition 3.3, whereas those depicted on the left of Fig. 2 and in Fig. 3 are not included in the above setting. Let us note that, although still not covering all realistic thin-film deformations, the set of approximately injective deformations encompasses a wider range of scenarios compared to those allowed by (2.10).

We provide below a construction of a recovery sequence for triples \((y,b,{\mathscr {M}})\in {\mathcal {A}}\) under the assumption that \(y\in {\mathcal {Y}}\).

In order to give a full \(\varGamma \)-convergence result, in the remainder of the section we restrict our analysis to deformations satisfying the following uniform averaged invertibility constraint: there exists a constant \(C>0\) such thatfor every \(x',\,z'\in \omega \). Note that the condition above has a pointwise meaning because maps with uniformly bounded energies are at least \(C^1\)-regular.

The key idea of (3.13) is that deformed vertical fibers might intersect, but are, in average, distant enough, compared to the distance of the original points in the cross section.

Let us start by remarking that, under the same assumptions of Proposition 3.2, and assuming additionally (3.13), the limiting deformations \(y\in W^{2,p}(\omega ;{\mathbb {R}}^3)\) have the additional property:In fact, property (3.14) follows from (3.5) and (3.13). In view of (3.14), we are in the position of obtaining the following \(\varGamma \)-convergence result.

A proof of the statement is in Sect. 4 below.

We collect in this section the proofs of the statements from Sect. 3. Within each subsection, notations are taken from the corresponding statement.