Magnetic control of magnetic shape-memory single crystals

Physica B 407 (2012) 1316–1321

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Magnetic control of magnetic shape-memory single crystals Ulisse Stefanelli a,b, a b

Istituto di Matematica Applicata e Tecnologie Informatiche – CNR, via Ferrata 1, I-27100 Pavia, Italy Weierstraß-Institut f¨ ur Angewandte Analysis und Stochastik, Mohrenstrasse 39, D-10117 Berlin, Germany

a r t i c l e i n f o

a b s t r a c t

Available online 29 June 2011

We present a phenomenological model for the magneto-mechanical evolution of shape-memory alloy single crystals. The existence of solutions for given magnetic field is commented and optimal control results are established. & 2011 Elsevier B.V. All rights reserved.

1. Introduction Shape-memory alloys (SMAs) are active materials: comparably large strains can be induced by either thermal or mechanical stimuli [21]. Some of these alloys (Ni2MnGa, NiMnInCo, NiFeGaCo, FePt, FePd, among others) are referred to as magnetic shape-memory alloys (MSMAs) as they feature a specific ferromagnetic character which entails a remarkable magnetostrictive behavior. For instance, a Ni2MnGa single crystal can develop up to a 10% strain (at a 1–3 MPa activation stress under the effect of a 1 T magnetic field) whereas a TerFeNOL-D polycrystal, one of the most performing giant magnetostrictive materials, shows a maximal 0.2% strain (at 60 MPa stress and 0.2 T field). The magnetically induced strains in MSMAs are the macroscopic effect of the orientation of the ferromagnetic martensitic variants of the material. In particular, the martensitic phase in MSMAs presents the classical ferromagnetic texture of magnetic domains. This mesostructure changes under the influence of an external field by magnetic-domain wall motion, magnetization vector rotation, and magnetic field driven martensitic-variant reorientation. The first two effects above are present in all ferromagnetic materials whereas martensitic-variant reorientation is specific of MSMAs. The engineering literature on MSMAs is already quite vast. The reader shall be referred, with no claim of completeness, to Refs. [15,24,25,31,40,45], see also the review in Ref. [26]. We shall be describing the microscopic martensitic phasefraction distribution of a MSMA single crystal by the vector p A Rv taking values in the simplex S :¼ fpi Z0, p1 þ    þpv r 1g. In particular, p ¼ 0 stands for a purely austenitic specimen whereas p A fp1 þ    þpv ¼ 1g means pure martensite and p=ðp1 þ    þ pv Þ represents the local martensitic variants distribution. We have

 Correspondence address: Istituto di Matematica Applicata e Tecnologie Informatiche – CNR, via Ferrata 1, I-27100 Pavia, Italy. Tel.: þ 39 0382 548202. E-mail address: [email protected]

0921-4526/$ - see front matter & 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2011.06.043

specifically in mind the cases v¼3 and 6 which correspond to cubic-to-tetragonal (three variants) cubic-to-orthorombic (six variants) austenite–martensite systems. Within this frame, we might assume each martensitic variant to show a specific so-called easy axis of magnetization. In particular, we assume that the linear relation p/Ap for given A A Rdv gives the (directed) easy axis of the phase distribution p (and jApj ¼ 1 for pure phases). Additionally, the orientation of the variants with respect to the easy axis will be determined by the scalar a A ½1,1. Our theory will rely on the ansatz that the magnetization M on the material is given by M ¼ msat aAp,

ð1Þ

where msat 4 0 is the saturation magnetization. In particular, we assume that the magnetic anisotropy of the material is sufficiently strong so that the magnetization stays rigidly attached to the easy axis of the martensitic variants and no magnetization rotation occurs. This assumptions are in agreement with experiments on Ni2MnGa [40,45]. Still, the reader is referred to Ref. [11] for the analysis of a more general version of this model including magnetization rotations and to Ref. [44] for a related optimal control result. In order to describe the complex thermo-magneto-mechanical behavior of a MSMA single crystal we shall rely on the modelization in Refs. [1,2,10] which corresponds to an extension of the celebrated Souza–Auricchio model for SMAs [4–6,42]. The latter is formulated within the frame of generalized plasticity and is characterized by a remarkable simplicity (few easily identifiable material parameters suffice in order to the describe a full 3D situation) and a variational structure (which entails robustness with respect to approximations and discretizations). Extensions of the Souza–Auricchio model as well as analytical results have been obtained in Refs. [3,7–9,16–19, 22,29,30,35–37,34]. In particular, the present constitutive model for MSMAs single crystals has been proved to admit weak solutions when coupled with quasi-static mechanics [10,11]. The giant magnetostrictive behavior of MSMA gives the unprecedented possibility of activating SMA devices (sensors, actuator, etc.)

U. Stefanelli / Physica B 407 (2012) 1316–1321

at a distance by tuning an external magnetic field. The aim of this paper is that of providing a first optimal control result in this direction. Namely, after having recalled the basic features of the MSMA model from [1,2,10] in Section 2, we address an optimal control problem where the admissible control is a time-dependent imposed magnetic field and controllable quantities are the displacement u from the reference configuration and the phase vector p. We shall first recall some existence theory for the state problem in Section 3 and finally provide our optimal controllability statement and proof in Section 4.

We shall recall here the basic features of the modelization from [1,2,10]. In the following bold latin letters stand for vectors in Rd ðd ¼ 2,3Þ bold greek symbols are for Rdd tensors, and we use the standard notation for scalar and contraction products. The dd space of symmetric tensors is Rdd sym and Rdev denotes its subspace of deviatoric elements. Let u be the displacement of the body the reference configuration. Moving within the small-strain regime, the linearized strain eðuÞ ¼ ðru þ ru> Þ=2 is additively decomposed as ð2Þ

Here C (symmetric, positive definite) is the isotropic elasticity fourtensor (assumed to be constant for all variants), r A Rdd sym is the stress, and the linear map p/e0 ðpÞ A Rdd dev represents the stress-free configuration corresponding to the phase distribution p. In particular,

e0 ðpÞ :¼ ei0 pi

ð3Þ

(summation convention) where ei0 is the stress-free reference configuration of the i-th martensitic phase. In case v¼ 3 a standard choice for d¼3 is given by

e ei0 ¼ pLffiffiffi ðI3ðei  ei ÞÞ, 6

ð4Þ

where ei is the unit vector of the i-th axis and eL 40 represents the maximal strain modulus obtainable via martensitic-variant reorientation. Note that, from relation (3), for every p A S we have that je0 ðpÞj r eL . Given the magnetic field H and the absolute temperature T, we shall prescribe the Gibbs free energy density of the material as 1 h Gðr,H,T,p, aÞ :¼  r : C1 rr : e0 ðpÞ þ bðTÞje0 ðpÞj þ je0 ðpÞj2 2 2 1 2 þ IS ðpÞ þ a þ I½1,1 ðaÞm0 H  amsat Ap: 2d

From the choice (5) of the Gibbs energy we derive the constitutive relations (1) and (2) as well as

n A @p G ¼ r : @p e0 ðpÞbn @p je0 ðpÞj@p IS ðpÞ he0 ðpÞ : @p e0 ðpÞ þ m0 amsat HA,

g A @a G ¼ a=d@a I½1,1 ðaÞ þ m0 msat H  Ap,

ð6Þ ð7Þ

v

where n A R and g A R are the thermodynamic forces associated with the internal variables p and a, respectively (non-smooth but convex functions are subdifferentiated in the sense of Convex Analysis [12]). Note that we readily have that ð@p e0 ðpÞÞijk ¼ ðek0 Þij ,

2. Constitutive material model

eðuÞ ¼ C1 r þ e0 ðpÞ:

1317

e0 ðpÞ : ek0 je0 ðpÞj

ð@p je0 ðpÞjÞk ¼

for e0 ðpÞ a 0:

The evolution of the material is prescribed via a normality flow rule. We assume the behavior of p to be dissipative. In particular, we prescribe the von Mises-type yield function F : Rv -R: FðnÞ :¼ jnjR, where R 40 is the activation radius, and require p_ to satisfy the flow rule and the complementary conditions: p_ ¼ z_ @FðnÞ,

z_ Z0, F r 0, z_ F ¼ 0:

The latter can be reformulated by means of the dissipation _ :¼ Rjpj _ as function DðpÞ _ n A @DðpÞ:

ð8Þ

On the other hand, we assume that a does not dissipate, namely g ¼ 0. This is of course disputable as the dissipation in a is the basic dissipative mechanism in ferromagnetic materials. Our assumption is however justified at the experimental level where it has been observed that the dissipation in a is negligible with respect to that in p [14,27]. As a does not dissipate, it can be minimized out from the Gibbs energy (5) as

a ¼ pðdm0 msat H  ApÞ, where p stands for the projection on the interval ½1,1. In particular, by letting, for all r A R, Fmag ðrÞ :¼

1 min fðdm0 msat rÞ2 ,2jdm0 msat rj1g, 2d

we can write the material constitutive relation (6) þ (8) (now without a) as ð5Þ

The first line in Eq. (5) corresponds to classical linearized elastoplasticity whereas the second line is the specific hardening choice of the Souza–Auricchio model. In particular, T/bðTÞ Z 0 represents the critical yield stress for the austenite–martensite transition at temperature T, h4 0 is an isotropic hardening modulus, and IS is the indicator function of the simplex S, namely IS ðpÞ ¼ 0 if p A S and IS ðpÞ ¼ 1 otherwise. From now on, we turn our attention to the isothermal situation by fixing the temperature to some n value Tn and letting b ¼ bðT n Þ. We shall use the notation FSA ðpÞ for the whole second line in Eq. (5). The third and final line in Eq. (5) describes the magnetic behavior of the material. The term m0 H  amsat Ap is the classical Zeeman energy term, namely m0 H  M (see Eq. (1)) and m0 is the magnetic permeability of vacuum. Note that H stands here for the internal magnetic field. Namely, H results from the sum of the applied external field and the corresponding induced demagnetization field. The indicator function I½1,1 is constraining a to the interval ½1,1 and 1=d is a user-defined (dimensionalized in MPa) hardening parameter.

_ þ@FSA ðpÞ@p Fmag ðH  ApÞ 3 r : @p e0 ðpÞ: @DðpÞ

ð9Þ

Let us now collect some remarks on the constitutive model (9). At first, note that, as jApj ¼ 1 for pure phases, we readily have from Eq. (1) that the natural constraint jMjr jmsat Apj ¼ msat jApj r msat is fulfilled for all p A S. Second, one has to stress that in the purely martensitic phase p A fp1 þ . . . þ pv ¼ 1g, standard choices for e0 and A (see Eqs. (3) and (4) for v¼3) entail that the model presents the so-called pairwise magnetic compatibility of energy wells [15]. In particular, in Ref. [1] we check that for all 1 r i,j r v there exist vectors aij ,nij A Rd such that 1 2

ei0 ej0 ¼ ðaij  nij þ nij  aij Þ,

ð10Þ

Api Apj ¼ nij :

ð11Þ

Condition (10) ensures that there exists a nontrivial continuous deformation such that eðuÞ takes value in fei0 , ej0 g. In particular, aij ,nij are the two possible normals to the discontinuity surface of the strain. On the other hand, condition (11) asserts that the interfaces

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U. Stefanelli / Physica B 407 (2012) 1316–1321

with normal nij serve as pole-free surfaces of discontinuity of the magnetization. From the purely mechanical viewpoint one has however to observe that the proposed model does not include the description of compatibility constraints between martensitic variants and austenite. In other words, the assumption p A S is indeed a simplification as in the region fp1 þ    þ pv o1g some phase proportions p are not accessible to real materials (this remark does not apply to the purely martensitic situation p1 þ    þ pv ¼ 1). Finally, our model reproduces, although to some schematic extent, the blocking stress effect. Namely, no hard-axis magnetically induced martensitic reorientation appears above a prescribed (and small) stress threshold [10]. Let us mention that different phenomenological models of internal variable-type for MSMAs have been advanced by Hirsinger and Lexcellent [23] and Kiefer and Lagoudas [28]. Albeit basically informed by the same principles, these two models differ from the present one as they are essentially restricted to two dimensions (or two martensitic variants), assume the scalar local proportion of one martensitic variant with respect to the other as the relevant internal variable (whereas here we have a full vectorial description via p), and rely on a considerably more complex choice of the Gibbs energy. In particular, the referred models anisotropy is directly built in by means of the choice of specific anisotropic energy contributions. Before closing this section let us motivate our interest for single crystals modeling by remarking that MSMAs polycrystals, despite their relatively easier production process, have not been exploited yet in real devices. One can offer two possible motivations for this fact. First, a significant drop in the magnetostrictive behavior for polycrystals vs. single crystals is observed, see Ref. [13] for a discussion. This drop is probably the outcome of the relatively poor martensitic-variant structure of all MSMAs to date (tetragonal, orthorombic) and compatibility conditions for magnetizations at grain boundaries. Second, one shall observe that MSMA polycrystals of Ni2MnGa developed so far turned out to be extremely brittle [43].

We shall now assume to be given the internal magnetic field ðx,tÞ/Hðx,tÞ A Rd and solve the material constitutive relation (9) together with the quasi-static equilibrium system: ð12Þ

d

Here O  R is a Lipschitz-bounded, connected open set and f : O-Rd is a prescribed body force. Given GDir [ Gtr ¼ @O with GDir having positive surface measure, the system (12) is complemented by the boundary conditions: u ¼ 0 on GDir ,

rn ¼ g on Gtr ,

ð13Þ

where n stands for the outward normal to @O, and g : Gtr -Rd is a given surface traction. The Dirichlet datum for u can be assumed non-homogeneous as well. We shall define the state space U  P 3 ðu,pÞ as U :¼ fu A H1 ðO; Rd Þ : u ¼ 0 on GDir g and P ¼ L1 ðO; Rv Þ. The total load t/lðtÞ A U 0 (dual) is given by Z Z /lðtÞ,uS :¼ f  u þ g  u: O

ðFSA ðpÞFmag ðH  ApÞÞ/lðtÞ,uS:

O

Note that the energy contains an interfacial term where n 40 is a scale parameter. In particular, the occurrence of such term penalizes phase interfaces. However it still does not prevent p from possibly exhibiting jumps. This is a particularly desirable feature in connection with shape-memory alloys where fewatoms-thick phase structures arise at the mesoscopic level. From the mathematical viewpoint, the interfacial energy term bears also a crucial compactifying effect. Finally, the dissipation functional D : P  P-½0,1Þ is given by Z Dðp1 ,p2 Þ :¼ R jp1 p2 j: O

We are now in the position of presenting a weak formulation of the quasi-static evolution of system (9), (12) and (13) within the frame of energetic solutions a la Mielke [33,39]. In particular, we define the set of stable states at time t A ½0,T and field H : O-Rd as Sðt,HÞ :¼ fðu,pÞ A U  P : Eðt,u,p,HÞ o1, b ,p b ,HÞ þDðp, p bÞ Eðt,u,p,HÞ r Eðt, u b,p b Þ A U  Pg: 8ðu Hence, given the magnetic field H : O  ½0,T-Rd and the initial state ðu0 ,p0 Þ A U  P, an energetic solution is a trajectory t A ½0,T/ðuðtÞ,zðtÞÞ A U  P such that ðuð0Þ,zð0ÞÞ ¼ ðu0 ,z0 Þ, t/@t Eðt,uðtÞ,pðtÞ,HðtÞÞ is integrable, and for all t A ½0,T, we have the two conditions: Stability: ðuðtÞ,pðtÞÞ A Sðt,HðtÞÞ:

ð14Þ

Energy balance: Eðt,uðtÞ,pðtÞ,HðtÞÞ þDissD ðp,½0,tÞ ¼ Eð0,u0 ,p0 ,Hð0ÞÞ Z t @t Eðs,uðsÞ,pðsÞ,HðsÞÞds, þ

ð15Þ

0

where the total dissipation of the process on the time interval ½s,t D ½0,T be given by

3. Existence for the state problem

r  r þ f ¼ 0 in O:

Z þ

Gtr

We prescribe the energy functional E : ½0,T  U  P  L1 ðO; Rd Þð1,1 as Z 1 Eðt,u,p,HÞ :¼ CðeðuÞe0 ðpÞÞ2 þ n VarðpÞ 2 O

DissD ðp,½s,tÞ :¼ sup

N X

Dðpðti1 Þ,pðti ÞÞ,

i¼1

the sup being taken among all partitions fs ¼ t0 ot1 o    otN ¼ tg. We shall make the following assumptions on body force and traction: ðf ,gÞ A W 1,1 ð0,T; L2 ðO; Rd Þ  L2 ðGtr ; Rd ÞÞ:

ð16Þ

Our result on the state problem reads as follows. Theorem 1 (Existence for the state problem). Assuming (16), H A W 1,1 ð0,T; L1 ðO; Rd ÞÞ, and ðu0 ,p0 Þ A Sð0,Hð0ÞÞ, there exists an energetic solution ðu,pÞ of the state problem. We shall not provide here a full proof of the latter result as the argument basically follows from the by-now classical existence analysis for energetic solutions [33]. In particular, an energetic solution may be obtained via passage to the limit within an implicit discretization procedure. Namely, by letting f0 ¼ t0 o t1 o    o tN ¼ Tg be a given time-partition and defining ðu0 ,p0 Þ ¼ ðu0 ,p0 Þ, we shall be interested in solving the following incremental problems: ðui ,pi Þ A Arg minðEðti ,u,p,Hðti ÞÞ þDðpi1 ,pÞÞ for all i ¼ 1, . . . ,N, where the minimum is taken in U  P. The (interpolant in time of the) time-discrete solution fðui ,pi ÞgN i ¼ 0 can then be proved to converge to a continuous energetic solution. Note however that the general method of Ref. [20] needs here a

U. Stefanelli / Physica B 407 (2012) 1316–1321

slight adaptation as the power of external actions: Z _l,uS _ t/ F 0mag ðHðtÞ  ApÞHðtÞA/

ð17Þ

O

need not be uniformly continuous. This modification is already mentioned on an abstract level on Ref. [41] and has been detailed for this model in Ref. [10]. We shall however give some detail on an a priori estimate on energetic solutions in terms of the magnetic field H. Henceforth, C stands for a positive constant depending on data and may vary from line to line. From the energy balance (15) we have that for all t A ½0,T: Z Z 1 CðeðuðtÞÞe0 ðpðtÞÞÞ2 þ n VarðpðtÞÞ þ FSA ðpðtÞÞ þ DissD ðp,½0,tÞ 2 O O ¼ Eðt,uðtÞ,pðtÞ,HðtÞÞ þ DissD ðp,½0,tÞ þ

Z

Fmag ðHðtÞ  ApðtÞÞ þ /lðtÞ,uðtÞS

1319

In order to possibly find optimal controls we shall consider the following standard requirements. Compatibility of initial values and controls: ðu0 ,p0 Þ A Sð0,Hð0ÞÞ

8H A H:

ð20Þ

Compactness of controls: H is compact in W 1,1 ð0,T; L1 ðO; Rd ÞÞ:

ð21Þ

Lower semicontinuity of the cost functional: ðH n -H strongly in W 1,1 ð0,T; L1 ðO; Rd ÞÞ, ðun ,pn Þ A SolðH n Þ, ðun ,pn Þ-ðu,pÞ weak-star in L1 ð0,T; U  PÞÞ ) J ðu,p,HÞ r lim inf J ðun ,pn ,H n Þ: n-1

ð22Þ

O

Z tZ Z t _ A ¼ Eð0,u0 ,p0 ,Hð0ÞÞ F 0mag ðH  ApÞH /_l,uS 0 O 0 Z þ Fmag ðHðtÞ  ApðtÞÞ þ /lðtÞ,uðtÞS:

ð15Þ

ð18Þ

O

As Fmag is Lipschitz continuous and pðtÞ A S, the above right-hand side can be bounded for all Z 4 0, by  Z t C ZJuðtÞJ2U þ 1 þJHJW 1,1 ð0,T;L1 ðO;Rd ÞÞ þJlðtÞJ2U0 þ JlJU 0 JuJU :

Z

0

Hence, by choosing Z small enough and applying Korn’s inequality and Gronwall’s lemma in Ref. (18), we deduce that   sup JuðtÞJ2U þ JpðtÞJP þ VarðpðtÞÞ þ DissD ðp,½0,TÞ t A ½0,T

r Cð1 þJHJW 1,1 ð0,T;L1 ðO;Rd ÞÞ Þ:

ð19Þ

Before closing this section let us observe that, by letting the parameter d-0, one can rigorously prove that the present magnetic model reduces to the original non-magnetic Souza–Auricchio model [10]. This asymptotic limit argument can be ascertained via the G-convergence theory for rate-independent processes devised in Ref. [38].

4. Optimal control Theorem 1 ensures the energetic solvability of the quasi-static evolution problem for a given space- and time-dependent field H. We shall denote by SolðHÞ the set of all such energetic solutions. Now, let us assume to be able to control H in order to optimize a given cost functional. Note again that H is the internal magnetic field whereas some more natural control variable would be the external magnetic field instead. These two perspectives are indeed equivalent if we assume, given ðu,pÞ, to be able to reconstruct in closed form the demagnetization field at every time. This is clearly too optimistic for the demagnetization tensor can be explicitly computed just in a few specific geometric situation and one would be forced to consider the coupling with the Maxwell system instead. We shall address this perspective in Ref. [44]. On the contrary, here we stick to this simplification by having in mind the case of relatively small displacements. Given a set of admissible magnetic fields (controls) H  W 1,1 ð0,T; L1 ðO; Rd ÞÞ, the optimal control problem consists of the minimization of a given cost functional: J : L1 ð0,T; U  PÞ  H-ð1,1,

The compatibility condition in Eq. (20) was already presented in Ref. [41] and is just intended to ensure that the initial values are stable regardless of the choice of the control. In case all H A H share the same initial value (which is somehow natural in applications where Hð0Þ ¼ 0 is usually taken) the compatibility condition (20) reduces to the purely mechanical stability of the initial state. The compactness of H from Eq. (21) is here chosen just for the sake of simplicity. In particular it can be relaxed by asking extra coercivity on the functional J . The lower semicontinuity requirement in Eq. (22) is standard. Let us now provide a first illustration of a possible quadratic cost functional covered by this theory. Indeed, we could consider Z T Z T J ðu,p,HÞ ¼ juud j2 þ jppd j2 0

0

þ juðTÞuf j2 þ jpðTÞpf j2 , where ðud ,pd Þ A L2 ð0,T; L2 ðO; Rd  Rv ÞÞ are given displacement and phase distribution profiles whereas ðuf ,pf Þ A L2 ðO; Rd  Rv Þ are given target states. Note that the latter functional is not lower semicontinuous with respect to the weak-star topology in L1 ð0,T; U  PÞ. Still, it fulfills Eq. (22) as a result of the requirement ðun ,pn Þ A SolðH n Þ which indeed provides additional compactness. In particular, as a consequence of the strong convergence H k -H in W 1,1 ð0,T; L1 ðO; Rd ÞÞ one has that (a suitable subsequence of) the corresponding solutions ðuk ,pk Þ A SolðH k Þ weakly star converge pointwise in U  P. Our optimal controllability statement reads as follows. Theorem 2 (Existence of optimal controls). Under assumptions (16), (20)–(22) there exists an optimal control. We provide here a direct proof of this theorem. Still, one shall mention that the result could be derived as a consequence of the abstract theory by Rindler [41] as well. Proof. Let ðuk ,pk ,H k Þ be a minimizing sequence for the functional J . Namely ðuk ,pk Þ A SolðH k Þ and J ðuk ,pk ,H k Þ-inf fJ ðu,p,HÞ j ðu,pÞ A SolðHÞg: Owing to the compactness (20) we extract a (not relabeled) subsequence such that H k -H n strongly in W 1,1 ð0,T; L1 ðO; Rd ÞÞ. Now, taking into account the estimate (19), one has that ðuk ,pk Þ are uniformly bounded in U  P and that supk DissD ðpk ,½0,TÞ o1. Hence, by Helly’s selection theorem [32] we can extract again (still not relabeling) in such a way that

which is depending on both the energetic solution and the control. Our problem is to find an optimal control H n A H and a corresponding optimal energetic solution ðun ,pn Þ A SolðH n Þ such that

pk ðtÞ-pn ðtÞ

ðun ,pn Þ A Arg MinfJ ðu,p,HÞ, such that ðu,pÞ A SolðHÞ, H A Hg:

pk -pn

weakly in BV ðO; Rv Þ

and strongly in L1 ðO; Rv Þ, strongly in Lq ðO  ð0,TÞÞ 8q A ½1,1Þ,

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U. Stefanelli / Physica B 407 (2012) 1316–1321

weakly in H1 ðO; Rd Þ:

uk ðtÞ-un ðtÞ

immediate as

In particular, the pointwise convergence of uk is obtained by stability (14) via the quadratic character of u/Eðt,u,p,HÞ. In fact, there exists a linear operator LðtÞ : L2 ðO; Rv Þ-H1 ðO; Rd Þ (independently of k) such that uk ðtÞ ¼ LðtÞpk ðtÞ. In particular, ðuk ,pk Þ-ðun ,pn Þ weakly star in L1 ð0,T; U  PÞ so that, by the lower semicontinuity assumption (22), we have that J ðun ,pn ,H n Þ r lim inf J ðuk ,pk ,H k Þ: k-1

M X

/lj lj1 ,ujn S ¼

Eðt,un ðtÞ,pn ðtÞ,H n ðtÞÞ þ DissD ðpn ,½0,tÞ r lim inf ðEðt,uk ðtÞ,pk ðtÞ,H k ðtÞÞ þ DissD ðpk ,½0,tÞÞ k-1

 ¼ lim inf Eð0,u0 ,p0 ,H k ð0ÞÞ k-1

Z

t

 0

/_l,uk S

Z

t

Z

0

O

¼ Eð0,u0 ,p0 ,H n ð0ÞÞ Z

t

Z

t

 O



/_l,un S

_ n  Ap , F 0mag ðH n  Apn ÞH n

j

j

j1

j

j

j1

Z Eðs ,un ,pn ,H n ÞEðs ,un ,pn ,H n Þ Z  Apjn Þ, ¼ /lj lj1 ,ujn S Fmag ðH jn  Apjn Þ þ Fmag ðH j1 n

/_l,un S:

0

Z TZ O

1

0

Z

1 0

! Z

F 0mag ðH jy  Apjn Þdy

sj

! _ n  Apj H n

sj1

! _ n  Ap , F 0mag ðH y  Ap n Þdy H n

ð24Þ

where H jy ¼ yH jn þ ð1yÞH j1 n . Let us check that ! Z 1 F 0mag ðH y  Ap n Þdy -F 0mag ðH n  Apn Þ 0

pointwise almost everywhere in space-time. Indeed, for t A ðsj1 ,sj  one has that  ! Z  Z 1    F 0mag ðH y  Ap n Þdy F 0mag ðH n  Apn Þ    O 0 Z Z 1  F 0mag ðH y  Ap n ÞF 0mag ðH n  Ap n Þ ¼  O 0   þF 0mag ðH n  Ap n ÞF 0mag ðH n  Apn Þdy Z Z sj JH_ n JL1 þC jH n Jp n pn j-0 rC O

by the Lipschitz continuity of F 0mag and the absolute continuity of H n . In particular, the space-time integrands in the last term of Eq. (24) _ n  Ap . converge pointwise almost everywhere to F 0mag ðH n  Apn ÞH n Hence, by Dominated Convergence, the respective integrals also converge. Eventually, the right-hand side of Eq. (23) converges to Z T Z TZ _ n  Ap  F 0mag ðH n  Apn ÞH /_l,un S  n 0

O

0

and the lower energy estimate and thus Eq. (15) follows. Hence, we have checked that ðun ,pn Þ A SolðH n Þ This concludes the proof. &

Acknowledgments

References

O

so that, by taking the sum for j ¼ 1, . . . ,M, one obtains that Eðt,un ðtÞ,pn ðtÞ,H n ðtÞÞEðs,un ðsÞ,pn ðsÞ,H n ðsÞÞ þ DissD ðpn ,½s,tÞ M Z  M  X X Fmag ðH jn  Apjn ÞFmag ðH j1  Apjn Þ  /lj lj1 ,ujn S: Z n j¼1 O

T

Partial support by the grants FP7-IDEAS-ERC-StG #200497 ˇ BioSMA, CNR-AVCR SmartMath, and by the Alexander von Humboldt Foundation is acknowledged.

j1 j1 j j1 Eðsj ,ujn ,pjn ,H jn ÞEðsj1 ,uj1 n ,pn ,H n Þ þ Dðpn ,pn Þ j

Z

sj1

so that the upper energy estimate holds. In the latter we have exploited the pointwise almost everywhere convergence F 0mag ðH k  _ k  Ap -F 0 ðH n  H _ n  Ap ÞAp (recall that F 0 Apk ÞH k n n mag mag is Lipschitz continuous) and Dominated Convergence. We shall now check the lower energy estimate. To this end fix ½s,t D½0,T, a partition fs ¼ s0 o s1 o    o sM ¼ tg, define ðujn ,pjn ,H jn , lj Þ ¼ ðun ðsj Þ,pn ðsj Þ,H n ðsj Þ,lðsj ÞÞ, and let u n , p n , etc. be the corresponding piecewise-constant interpolants on the partition. Note in particular that u n -un weakly star in L1 ð0,T; UÞ and for all t A ½0,T, p n ðtÞ-pn ðtÞ strongly in Lq ðO; Rv Þ for all q A ½1,1Þ. By j1 j1 ,H j1 exploiting ðuj1 n ,pn Þ A Sðs n Þ we have

j

M Z X j¼1 O

0

Z

0

_ k  Ap F 0mag ðH k  Apk ÞH k

¼

0

where we have used the lower semicontinuity of E and the continuity of D in ðL1 ðO; Rv ÞÞ2 . Hence ðun ðtÞ,pn ðtÞÞ A Sðt,H n ðtÞÞ. By passing to the lim inf in the energy balance (15) we obtain that

Z

O

¼

b Þ, b ,p b ,H n ðtÞÞ þDðpn ðtÞ, p ¼ Eðt, u

/_l,u n S-

As for the remainder term in the right-hand side of Eq. (23) we argue as follows: M Z X ðFmag ðH jn  Apjn ÞFmag ðH j1  Apjn ÞÞ n j¼1

k-1

T 0

j¼1

In order to conclude the proof we now aim at showing that ðun ,pn Þ A SolðH n Þ. Let us start from checking stability (14). For all b ,p b Þ A U  P we have given t A ½0,T and ðu Eðt,un ðtÞ,pn ðtÞ,H n ðtÞÞ r lim inf ðEðt,uk ðtÞ,pk ðtÞ,H k ðtÞÞ k-1   b,p b ,H k ðtÞÞþ Dðpk ðtÞ, p bÞ rlim inf Eðt, u

Z

j¼1

ð23Þ In order to conclude for the lower energy estimate one has to check that the above right-hand side converges to the integral on ½s,t of the power of external actions (17) as the diameter of the given partition goes to 0. The treatment of the loading term is

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