Boundary conditions, RVE size and microstructure composition Aspects of computational homogenization in magneto-mechanics: International Journal of Solids and Structures

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International Journal of Solids and Structures

journalhomepage:www.elsevier.com/locate/ijsolstr

Aspects of computational homogenization in magneto-mechanics:

Boundary conditions, RVE size and microstructure composition

R. Zabihyan

^a^,^∗

, J. Mergheim

^a

, A. Javili

^b

, P. Steinmann

^a

a Chair of Applied Mechanics, Universität Erlangen–Nürnberg, Egerlandstr. 5, Erlangen 91058, Germany

b Department of Mechanical Engineering, Bilkent University, Ankara 06800, Turkey

a rt i c l e i n f o

Article history:

Received 29 May 2017 Revised 26 September 2017 Available online 17 October 2017 Keywords:

Homogenization Magneto-mechanics Boundary condition

a b s t r a c t

Inthe presentwork, thebehavior ofheterogeneousmagnetorheological compositessubjectedtolarge deformationsand externalmagneticﬁeldsisstudied.Computationalhomogenizationisused toderive themacroscopicmaterialresponsefromtheaveragedresponseoftheunderlyingmicrostructure.Themi- crostructureconsistsoftwomaterialsandisfarsmallerthanthecharacteristiclengthofthemacroscopic problem.Differenttypesofboundaryconditionsbasedontheprimaryvariablesofthemagneto-elastic enthalpyandinternalenergyfunctionalsareappliedtosolvetheproblematthemicro-scale.Theover- allresponsesoftheRVEswithdifferentsizesandparticledistributionsarestudiedunderdifferentloads andmagneticﬁelds.Theresultsindicatethattheapplicationofeachsetofboundaryconditionspresents differentmacroscopicresponses.However,increasingthesizeoftheRVE,solutionsfromdifferentbound- aryconditionsgetclosertoeachotherandconvergetotheresponse obtainedfromperiodicboundary conditions.

1. Introduction

Magneto-active elastomers consist of a soft polymer matrix filled withmagneto-active particles. Theychangetheir properties with the application of a magnetic field. The non-linear elastic characteristics of the matrix and the magnetic properties of the particles enable them to undergo very large and adjustable deformations in responseto relatively low external magnetic fields.

Due to such speciﬁcations thesematerials are of special interest for several engineering applications and attracted signiﬁcant re- search attention (Kordonsky, 1993; Jollyet al., 1996; Carlson and Jolly,2000).

Thetheoreticalaspects ofthemagneto-mechanicalresponseof solidshavebeenthoroughlystudiedinthepast.Thegeneralequa- tionsofmagneto-elasticityandthesolutionoftheresultingbound- aryvalueproblemshavebeenconsideredintheliterature(Kovetz, 2000;Vu andSteinmann, 2007;Bustamante etal., 2008; Vu and Steinmann, 2010;Bustamante et al., 2011). In particular, the de- velopment ofconstitutivelaws formagnetorheological elastomers areconsideredinBrigadnovandDorfmann(2003),Kankanalaand Triantafyllidis (2004), Steigmann (2004), Dorfmann and Og-

∗ Corresponding author.

E-mail addresses: reza.zabihyan@ltm.uni-erlangen.de , reza.zabihyan@fau.de (R.

Zabihyan), julia.mergheim@ltm.uni-erlangen.de (J. Mergheim), ajavili@bilkent.edu.tr (A. Javili), paul.steinmann@ltm.uni-erlangen.de (P. Steinmann).

den (2004) and Danas et al. (2012), among others, where the magnetorheological composites have been modeled on a macroscopic level.However, the strong dependency ofthe material re- sponseofthecomposites ontheirmicrostructure (e.g.shape,distribution, volume fraction and orientation of the particles) re- veals the importance of multi-scale modeling techniques, where the macroscopic response is determined from the response of the material microstructure. Comprehensive reviews of various multi-scale modeling techniques are given in Saeb et al. (2016), Pinderaetal.(2009)andMatousetal.(2017).

Oneofthemostwidelyusedmulti-scaletoolstopredictthebe- haviorofinhomogeneousmaterials iscomputationalhomogeniza- tion which isbased on Hill (1963) andHill andRice (1972) and allows to study the effective behavior of composite materials (Kouznetsovaetal.,2001;MieheandKoch,2002;ZohdiandWrig- gers, 2001). The modeling of the behavior of composites based on computational homogenization in the context of the small strain as well as the ﬁnite strain setting has been studied ex- tensively (Terada andKikuchi, 1995; Ponte Castañeda, 1996; Ter- adaand Kikuchi, 2001; Yvonnet et al., 2009; Miehe etal., 1999;

Kouznetsova et al., 2002; Costanzo et al., 2005; Hirschberger et al., 2008; Temizer and Wriggers, 2008; Javili et al., 2013b), amongothers.Veryrecently,homogenizationtechniqueshavebeen utilized to study multiphysics problems. Extensions to coupled electro-mechanicalresponsehavebeenaddressedinSchröderand Keip (2012), Kuznetsov and Fish (2012), Castañeda and Si- https://doi.org/10.1016/j.ijsolstr.2017.10.009

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boni (2012) and Keip et al. (2014) and thermo-mechanical problems have been studied in Özdemir et al. (2008) and Temizerand Wriggers(2011). Moreover, magneto-mechanicalhomogenization of magnetorheological elastomers is considered in Borcea and Bruno (2001), Wang et al. (2003), Yin et al. (2006), PonteCastañedaandGalipeau(2011)andGalipeauandPonteCas- tañeda (2013). In the field of magneto-elasticity for finite deformations, Castañeda and Galipeau (2011), Ponte Castañeda and Galipeau (2011) and Galipeau andPonte Castañeda (2013) intro- duceda finite-strain variational formulation where magnetoelas- tic effects are handled by means of the deformation-dependent magnetic susceptibility of the material. Danas (2017) proposed an augmented vector potential variational formulation to carry out numerical periodic homogenization studies on the mag- netoelastic composites at finite strains and magnetic fields.

Chatzigeorgiou et al. (2014) presented a general homogenization frameworkformagnetorheologicalelastomersunderﬁnitestrains.

They showed that the use of kinematic and magnetic field potentials,i.e. the variables of the enthalpy formulation, instead of kinetic field and magnetic induction potentials provides a more appropriate homogenization framework and convenient numerical implementation procedure. Based on this result, the numerical implementation of the homogenization procedure in finite strain magneto-mechanics is studied in Javili et al. (2013a) and KeipandRambausek(2016).However,Mieheetal.(2016)showed thatthecomputationalframeworkbasedonthesaddle-point-type magneto-elastic enthalpy functional is not advisable in order to detect instability points which occur in magnetorheological elastomers.¹ On the other hand, compared to the internal energy- based computational framework, the enthalpy-based formulation is very convenient fornumerical implementation due to the re- ductionofthemagneticvector potential toamagneticscalar potential. Therefore they proposed an internal energy-based computational homogenizationframework based on scalar potentials byreformulation ofthe energyinterms ofan averaged enthalpy functional. In the similar context, Gil and Ortigosa (2016) pro- poseda convexmulti-variable framework for the analysis ofthe electro-active polymers undergoing large deformations and electric fields which satisfies material stability for the entire range of deformations and electric fields. These considerations of the internal energy density functional in terms of a convex multi- variablefunctionofelectromechanicalargumentsareinevitablein problems where the electro-mechanical enthalpy based formula- tionfailsandyieldsnon-physicalmaterialresponses(Ortigosaand Gil, 2016b). Using the same analogy, the behavior of the incom- pressible electro-active polymers and electro-active shells under- goinglarge deformations andelectricfieldshavebeen studied in Ortigosaetal.(2016) andOrtigosa andGil. (2017).The extension ofthevariationalconvexmulti-variableframeworkfortheanalysis ofelectro-magneto-mechanical internal energyfunctionals iscar- riedoninOrtigosaandGil(2016a)wherethematerialstabilityof convexandnon-convexmulti-variableconstitutivemodelsisstud- ied.

Central to computational homogenization is the Hill–Mandel conditionwhichhastobesatisfiedbychoosingappropriatebound- aryconditions forthe microscopic problem. In the field of small strains Borcea and Bruno (2001) considered several types of boundary conditions and used prescribed displacements or tractions andan applied magnetic field at the boundary ofthe RVE.

The application ofa magnetic ﬁeld as a boundary condition has also been considered in other micro-scale models together with

1 The stability analysis of the enthalpy based framework can be carried out using complex arc-length methods where the control variable is a combination of the magnetic potential and the magnetic ﬁeld. See Belytschko et al. (2013) and Gil and Ortigosa (2016) for further details.

boundarytractions(Yinetal.,2006)andstrains(Yinetal.,2002).

The homogenizationof magnetostrictiveparticle-filled elastomers underperiodicboundaryconditionsandconstantmagnetostrictive eigen-deformationintheferromagneticparticleshasbeenstudied in Wangetal. (2003).Ponte Castañeda andGalipeau (2011) prescribed thedeformation gradient tensorandthe magneticinduction vector on the boundary ofthe microstructure andproposed anewhomogenizationframeworkformagneto-elastic composites which accounts forthe effect of magnetic dipoleinteractions, as wellasfinitestrains.Chatzigeorgiouetal.(2014)identifiedseveral casesofuniformboundaryconditionsontheRVEunderwhichthe Hill–Mandelconditionholds.

The current contribution isan extension to the theory devel- oped byChatzigeorgiou etal.(2014).We numericallyanalyzethe computationalmagneto-mechanicalhomogenizationframeworkin the finite deformation settingwith special focuson the two dif- ferentsets offormulations basedon theprimary variables ofthe magneto-elasticenthalpyW(^F,H)ând^themagneto-elasticinternal energyW^∗(^F,B) functionals. For each set, we investigateseveral combinationsofboundaryconditionsthat satisfy theHill–Mandel condition. Severalnumericalexamplesare giventostudythemi- croscopicand macroscopicresponses ofthe RVEs with magneto- mechanical constituents whichare differentin sizesand particle distributions. Furthermore, the influence of the boundary condi- tionsontheoverallresponseofvariousmicrostructures,underdif- ferentloadsandmagneticfields,arestudiedanddiscussedinde- tail.

The structure ofthe paper isas follows.In Section 2, we de- scribe the theoretical homogenization framework by presenting theﬁeldvariables,thebalanceequations,themagneto-mechanical constitutive model that accounts for large deformations and the scale transition.Different sets ofboundaryconditions thatsatisfy theHill–Mandelconditionaredescribed.InSection3,theresponse ofvariouscompositematerialswhichdifferinparticlesizeanddis- tributionare studiedundervarious magneto-mechanicalloadings.

Finally,Section4concludesthiswork.

2. Magneto-mechanicalhomogenization

The objectiveofthissection isto summarizecertain keycon- ceptsinthe coupledmagneto-mechanical homogenizationframe- workandnonlinear continuummechanics. Duetothe inhomoge- neousstructure ofcompositematerials, itis essentialto consider separately microandmacroscales. Themacro-scaledescribesthe continuum body andthemicro-scaledescribesthe representative volume element (RVE)of the microstructure.As it is depictedin Fig.1, both macro-scaleandmicro-scalecan be expressed inthe material or in the spatial conﬁguration. In the current work we formulatetheprobleminthematerialdescription.

2.1. Micro-problemdeﬁnition

IntheundeformedconﬁgurationB0atthemicro-scale,theRVE occupies thevolume V₀ with boundary

∂

B0 (Fig. 1) andconsists oftwomaterials. InthedeformedconﬁgurationBt the RVEoccu- piesthevolumeV_t withboundary

∂

Bt.Thenormalvectorstothe boundaries

∂

B0 and

∂

Bt are denoted N and n, respectively. The nonlineardeformationmapx=

φ

(^X)^describes^the^position^vector xof a point in the spatial conﬁgurationBt in terms of the position vector X of the point in the material conﬁguration B0. The microscopic deformationgradient Fisconnected tothe deforma- tionmap

φ

^through^the^relation

F=

∇

X

φ

. (1)

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Fig. 1. Macro-scale and micro-scale in the material and the spatial conﬁguration, enthalpy-based formulation.

Intheabsenceofinertiaandmechanicalbodyforcesatthemicro- scale,theconservationoflinearmomentumreads

DivP=0 inB0 . (2)

wherePisthePiolastress.Itconsistsofthesumofthemechanical stress, the Maxwell stress andthestress due tomaterial magne- tization, seee.g. Steigmann (2004). The divergenceoperator with respecttothematerialcoordinatesXisdenotedDiv•.

Ignoring any free current density at the micro-scale, the La- grangian magnetic ﬁeld H is connectedwith the scalarmagnetic potential intheundeformedconﬁguration

ϕ

(^X)^through^the^relation

H=

∇

^X

ϕ

^. ⁽³⁾

Moreover,theconservationofmagneticﬂuxiswrittenas

DivB=0 inB0, (4)

withthemagneticinductionB.Themicroscopicproblemhastobe completedby constitutiveequationsforthePiolastressP andthe magneticinductionBandboundaryconditionswhichfollowfrom thescale-transitionandwillbediscussedindetailinSection2.3.

2.1.1. Microscopicmagneto-mechanicalconstitutivemodel

With the help of magneto-elastic internal energy functionals, we can identify the constitutive relations that connect the Piola stress P andthemagneticinduction Bwiththedeformation gradient F and the magnetic ﬁeld H. Using the (saddle-point-type) magneto-elasticenthalpyfunctionW(^F,H)^the^Piola^stress^and^the Lagrangianmagneticinductionaregivenby

P=P

(

^F,H

)

=

∂

^W

(

^F,H

)

∂

^F ^, ^B⁼^B

⁽

^F^,^H

⁾

⁼⁻

∂

^W

(

^F,H

)

∂

H . (5) Using a Legendretransformation themagneto-elastic internal energy density function W^∗(^F,B), which is poly-convex in Fwhen

(B=0) andconvexin the magneticinduction B,see GilandOr- tigosa(2016),canbeidentiﬁedas

W^∗

(

^F,B

)

=sup

H [W

(

^F,H

)

+B· H]. (6)

Furthermore,theconstitutiverelationsforthePiolastressandthe magneticﬁeldcanbe derivedbasedonthemagneto-elasticinter- nalenergydensityfunctionW^∗(^F,B)^as

P=P

(

^F^,^B

)

⁼

∂

^W^∗

(

^F^,^B

)

∂

^F ^, ^H⁼^H

⁽

^F^,^B

⁾

⁼

∂

^W^∗

(

^F^,^B

)

∂

^B ^. ⁽⁷⁾

We assume an isotropic elastic material and an isotropic linear magneticresponse forthematrixmaterial andthemagneticpar- ticles, respectively. In order to establish appropriate constitutive relations and a convenient numerical implementation, the for- mulations are ﬁrstly derived in terms of a magneto-elastic en- thalpyfunctionalW(^F,H),whichforthesakeofdemonstrationis herebasedonNeo-Hookean hyperelasticity, compare(Javilietal., 2013a)

W

(

^F^,^H

)

⁼ ¹₂

λ

1[F:F− Dim− 2lnJ]+1 2

λ

2ln² J

−1

2

μ

^JH· C^-¹· H, (8)

where

λ

1and

λ

2 aretheLamé parameters,

μ

^is^the^magnetic^per-

meability andDim is the problem dimension. Also, C=F^t· Fde- notestherightCauchy–Green deformationtensorandJ=det(^F) .²Fromtheconstitutiverelation(5)thePiolastressPreads P=

∂

^W

(

^F,H

)

∂

^F ⁼

λ

¹^F⁺^[

λ

²^ln^J⁻

λ

¹^]^F^−t⁻¹₂

μ

^J^HH:M, (9) withMdeﬁnedas

M:=C^-¹F^−t+

∂

^C^-¹

∂

^F ^. ⁽¹⁰⁾

Furthermore, the magnetic induction B can be computed from (5)as

B=−

∂

^W

(

^F,H

)

∂

H =

μ

^J^C^-¹· H. (11)

As can be seen in Fig. 2(a) the enthalpy is a saddle-point-type function whichdoes not permit ina straightforward manner the solutionof stabilityproblemsorthe derivation ofboundson the homogenizedsolution.Aquasi-convexmagneto-elasticinternalen- ergy densityfunctional, which isalso shown inFig. 2(b),can be representedusingEq.(6)as

W^∗

(

^F,B

)

=sup

H [W

(

^F,H

)

+B· H]

=1

2

λ

¹^[^F^:^F^{− Dim}^{− 2}^ln^J^]⁺¹₂

λ

²^ln²^J⁺¹₂

μ

¹^J^B^{· C}^{· B}^.

(12) FromEq.(7)thePiolastressPthenreads

P=

∂

^W^∗

(

^F,B

)

∂

^F ⁼

λ

1F+[

λ

2lnJ−

λ

1]F^−t+ 1

μ

^J^F^{· [}^B^B^]

− 1

2

μ

^J^[^B^{· C}^{· B}^]^F^−t^. ⁽¹³⁾

Moreover,themagneticﬁeldHisderivedfrom(7)as H=

∂

^W^∗

(

^F,B

)

∂

B = 1

μ

^J^C^{· B}^. ⁽¹⁴⁾

Notethat(9)and(13)aswell as(11)and(14)areequivalentex- pressions,howeverindifferentparameterization.

2 It should be mentioned that for very large volumetric strains the strain energy functional (8) does not fulﬁll the polyconvexity condition in F when ( H = 0 ), i.e.

∂JJ W ( F ) ≥ 0, see Doll and Schweizerhof (1999) . However, in our study the volumetric strain is always small and therefore the polyconvexity condition is not violated.

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Fig. 2. Plots of (left) the macroscopic saddle-point magneto-elastic enthalpy W (F , H ) and (right) the macroscopic magneto-elastic internal energy W ^∗(F , B ) functionals with respect to [ F ] xx , [ H ] x and [ B ] x .

2.2.Macro-problemdeﬁnition

A macroscopiccontinuumbody occupiesthematerial conﬁgu- rationB0withboundary

∂

B0andthespatialconﬁgurationBt with boundary

∂

Bt, seeFig. 1. Forstationary applications,the macro- scopicequilibriumequation in thematerial conﬁgurationiswrit- tenas

DivP+b0=0 inB0 subjectto

φ

⁼

φ

^p ^on

∂

^B⁰^D^mech ^and

[[T]]=[[P· N]]=T^p on

∂

^B⁰^N^mech^, ⁽¹⁵⁾

with b₀ and P are the macroscopic body force density and the macroscopicPiolastressinthematerialconﬁguration,respectively.

φ

^and^T^deﬁne^themacroscopicdeformation andtractions.More- over,

φ

^p ^is^the^prescribed deformation onthe mechanicalDirich- letboundaryandT^pdenotesthemacroscopicprescribedtractions on the mechanical Neumann boundary, with

∂

B0=

∂

B0D_mech

∂

B0Nmechand

∂

B0Dmech∩

∂

B0Nmech=∅. ∪

Theconservationofthemacroscopicmagneticﬂuxiswritten

DivB=0 inB0 subjectto

ϕ

⁼

ϕ

^p ^on

∂

^B⁰^D^mag ^and

[[T]]=[[B· N]]=T^p on

∂

^B⁰^N^mag^. ⁽¹⁶⁾

where T^p is the prescribed macroscopic magnetic ﬂux on the magneticNeumannboundary

∂

B0Nmag.Themacroscopicprescribed magneticpotentialisdescribedwith

ϕ

^p^.

Due to the complex microstructure of composite materials, theconstitutiverelationsbetweenmacroscopicstress,macroscopic deformation gradient, the macroscopic magnetic induction and macroscopicmagnetic ﬁeld are not explicitly expressed here, instead they are determined by means of computational homoge- nizationi.e.usingthesolutions oftheRVEproblematthemicro- scale. The focus of the current work is on the microscopic re- sponseoftheRVEswhilestudyingthemacro-scalesolutionisout of the scope of this paper. For complete two-scale solutions for magneto-active composites see e.g. Javili et al. (2013a),Keip and Rambausek(2016)andSridharetal.(2016).

2.3.Micro-to-macrotransition

Inordertoestablishaconsistenttransitionbetweenthemicro- and macro-scale, the Hill–Mandel condition is required to be satisﬁed which stipulates the equivalence of the macroscopic variational enthalpy and the averaged microscopic variational enthalpy³

3 The Hill–Mandel condition could also be formulated in form of the variational magneto-elastic internal energy density as δW ^∗= F : δP + H ·δB which results in the same boundary conditions and is therefore not described.

1 V0

B0

δ

^W ^d^V⁻

δ

^W

= 1 V0

B0

[P:

δ

^F+B·

δ

H]dV− [P:

δ

^F+B·

δ

H]=0. (17)

ThemechanicalandthemagnetictermsoftheHill–Mandelcondi- tioncanbeseparatedas

1 V0

B0

P:

δ

^F^d^V^{− P}^:

δ

^F⁼⁰^, ^(18a)

1 V0

B0

B·

δ

^H^d^V^{− B}^·

δ

^H⁼⁰^, ^(18b)

respectively.

2.3.1. Mechanicalaveragevariablesandboundaryconditions We assume that the microscopic motion

φ

^is ^linked ^to ^the

macroscopic deformation gradient by the standard ﬁrst-order ansatz

φ

=F· X +

φ

(^X),where

φ

îs^the^vector^fluctuation^field.Ûs-

ing Eq.(1)yields themicroscopic deformationgradient F=F+F, withthegradientoftheﬂuctuationsF=

∇

X

φ

(^X)^.Substitutingthis into(18a)resultsin

1 V₀

B0

PdV− P

:

δ

^F+

1 V₀

B0

P:

δ

^F^d^V

=0. (19)

TheﬁrstterminEq.(19)vanishesifthemacroscopicPiolastressis equaltothevolumeaverageofitsmicro-scalecounterpart,i.e.

P= 1 V0

B0

PdV. (20)

Thesecondtermin(19)isreformulatedasaboundaryintegral 1

V0

B0

P:

δ

^F^d^V= 1 V0

∂B0

δ

φ

· TdA=0 with

δ

φ

=

δφ

− F· X, (21)

andbecomeszerobyimposingoneofthefollowingconstraintson theﬂuctuationﬁeld:

• Voigt’sassumption

φ

=F· X inB0 ,

• Lineardeformations

φ

=F· X on

∂

B0 (²²)

• Periodicdeformations

φ

⁺=

φ

⁻ ^and

anti-periodictractions T⁺=−T⁻ on

∂

B0.

TheNeumann-typeconstraintscanbederivedbyaprioriassuming anadditivedecompositionofthePiolastressintomacroscopicand ﬂuctuatingparts,P=P+P.Substitutingthisresultin(18a)gives

1 V₀

B0

[P+P]:

δ

^F^d^V^{− P}^:

δ

^F⁼^P^:

1 V₀

B0

δ

^F^d^V⁻

δ

^F

(5)

+1 V0

B0

P:

δ

^F^d^V ⁼⁰^. ⁽²³⁾

Itfollowsthatthemacroscopicdeformationgradientisthevolume averageofthemicroscopicdeformationgradient

F= 1 V0

B0

FdV, (24)

and that the ﬂuctuation termof the relation (23)should vanish.

UsingthedivergencetheoremandEq.(2)yields theboundaryin- tegral

1 V0

B0

P:

δ

^F^d^V ⁼_V¹

0

B0

Div

( δφ

^·^P

)

^d^V

= 1 V0

∂B0

δφ

· P· NdA=0. (25) whichvanishesifoneofthefollowingconstraintsissatisﬁed:

• Reuss’assumption P=P inB0.,

• Constanttractions P· N=P· N on

∂

B0 (²⁶) 2.3.2. Magneticaveragevariablesandboundaryconditions

In a similar fashion to the mechanicalproblem, we assume a linear ﬁrst-order ansatz for the microscopic magnetic potential,

ϕ

=H· X+

ϕ

(^X),where

ϕ

îs^the^scalar^fluctuation^field.^Therefore, Eqs.(3)yieldsthemicroscopic magneticfieldH=H+Hwiththe fluctuation term H=

∇

X

ϕ

(^X)^. Considering these equations and (18b)theHill–Mandelconditioncanbewrittenas

1 V₀

B0

BdV− B

·

δ

^H⁺

1 V₀

B0

B·

δ

^H^d^V

=0, (27)

whereby the ﬁrst term vanishes if the macroscopicmagnetic in- ductionisequaltoitsaveragedmicroscopiccounterpart

B= 1 V0

B0

BdV , (28)

andthesecondtermisformulatedasaboundaryintegral 1

V0

B0

B·

δ

^H^d^V⁼_V¹

0

∂B0

δ ϕ

^{· T}^d^A ⁼⁰ ^with

δ ϕ

⁼

δϕ

^{− H}^{· X}^. ⁽²⁹⁾

Eq. (29) can be satisﬁed by imposing one of the following con- straintsonthemicroscopicmagneticﬁeld:

• Voigt’sassumption

ϕ

=H· X inB0

• Linearmagneticpotential

ϕ

=H· X on

∂

B0 (³⁰)

• Periodicmagneticpotential

ϕ

⁺=

ϕ

⁻^and anti-periodicmagneticﬂux T⁺=−T⁻ on

∂

B0.

TheNeumann-typemagneticconstraintscanbeobtainedbyapri- oriassuminganadditivedecompositionofthemagneticinduction into macroscopic and ﬂuctuating parts B=B+B. Inserting this deﬁnitioninto(18b)gives

1 V0

B0

[B+B]·

δ

^H^d^V^{− B}^·

δ

^H⁼^B^·

1 V0

B0

δ

^H^d^V⁻

δ

^H

+ 1 V₀

B0

B·

δ

HdV =0. (31)

SatisfactionoftheHill–Mandelconditionrequiresthatthemacro- scopic magnetic ﬁeld is the volume average of the microscopic magneticﬁeld

H= 1 V0

B0

HdV , (32)

and that the ﬂuctuation term of the relation(31) should vanish.

UsingthedivergencetheoremandEq.(4)yields aboundaryinte- gral

1 V₀

B0

B·

δ

^H^d^V ⁼_V¹

0

B0

Div

( δϕ

^·^B

)

^d^V

= 1 V0

∂B0

δϕ

^·^B^{· N}^d^A⁼⁰^, ⁽³³⁾

whichbecomeszeroontheboundary

∂

B0 ifoneofthefollowing constraintsissatisﬁed:

• Reuss’assumption B=B inB0,

• Constantmagneticinduction B· N=B· N on

∂

B0. (34) Table1summarizesdifferentcombinationsoftheboundarycondi- tions,derivedinEqs.(22),(26),(30),(34).Theﬁrstsetisbasedon the primary variables of the magneto-elastic enthalpy functional W(^F,H)^.^Thisformulationresultsinasaddle-pointproblemwhich isnotstraightforward toperforma classicalstabilityanalysisand alsoisnotcapableoftrackingpostcriticalsolutionpathsassociated withinstabilities.Therefore,stabilityanalysis shouldbe based on theenergeticformulation(Mieheetal.,2016).Thesecondgroupof boundaryconditionsinTable1arebasedontheprimaryvariables ofthemagneto-elasticinternalenergyfunctionalW^∗(^F,B)^.

Remark:NotethattheVoigt’sandReuss’assumptionsbasedon prescribed deformation gradient F andmagnetic ﬁeld H are dif- ferentfromtheVoigt^∗andReuss^∗assumptionsbasedonvariables oftheinternalenergyfunction-prescribeddeformationgradientF andmagneticinductionB-andtheyshallnotbemistaken.

The numerical implementation of the microscopic problem based on the ﬁnite element method and the algorithms to prescribe the various boundary conditions are described in AppendixA.

3. Numericalexamples

The objective of this section is to present numerical examples in order to study the inﬂuence of differentmicrostructures, boundary conditionsand RVE sizeson the macroscopicresponse ofmagneto-mechanicalcomposites.Thecompositesareconsidered inaplane-strainsettingandconsist ofamatrixmaterial andcir- cular inclusions. In the following numerical examples the material parameters of the matrix are assumed to be: Lamé parameters

λ

^mat.1 =8,

λ

^mat.2 =12, magnetic permeability

μ

^mat^.=0.001, andthe material parameters of the inclusion are: Lamé parameters

λ

^inc1 ^.=80,

λ

^inc2 ^.=120andmagneticpermeability

μ

^inc^.=0.01. Simple-extensionandsimpleshearloadsareprescribedbyimpos- ingthedeformationgradientinx-directionorinthexy-plane, F=

[F]xx 0

0 1

, F=

1 [F]xy

0 1

.

Furthermore,amagneticloadingisappliedbyimposingeitherthe magneticﬁeldorthemagneticinductioninx-direction

H=

[H]x

0

, B=

[B]x

0

.

Allexamplesaresolvedusingourin-houseﬁniteelementcodeand are discretized withbilinear quadrilateralQ₁ elements. The solu- tionprocedureisrobustandshowsforallexamplesasymptotically the quadratic rate of convergence associated with the Newton- Raphsonscheme.

3.1. Unitcellundermagneticﬁeld,stretchingandshearing

Theaimofthisﬁrstexampleistodemonstratetheinﬂuenceof differentboundaryconditionsontheeffectiveandthemicroscopic responseofthe RVE.Acompositeconsisting ofamatrix material andperiodicallyalignedinclusionsisrepresentedbymeansofthe two-dimensionalunitcellillustratedinFig.3.

Fig. 4 shows the distributions of the microscopic normalized Piola stress and the microscopic normalized magnetic induction

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Table 1

Conditions for satisfying the Hill–Mandel condition.

Constraint Conditions based on prescribed ( F , H )

Voigt Lin. displacement: φ= F · X Lin. magnetic potential: ϕ = H · X in B 0

LD-LP Lin. displacement: φ^{= F · X} Lin. magnetic potential: ϕ^{= H · X} ^on∂^B0 Alg. 1 PD-PP Per. displacements: φ⁺⁼φ⁻ Per. magnetic potential: ϕ⁺ = ϕ⁻ ^on∂^B0 Alg. 2 CT-CI Const. traction: P · N = P · N Const. magnetic induction: B · N = B · N on ∂B 0 Alg. 3 Reuss Const. traction: P = P Const. magnetic induction: B = B in B 0

Constraint Conditions based on prescribed ( F , B )

Voigt ^∗ Lin. displacement: φ= F · X Const. magnetic induction: B = B in B 0

LD-CI Lin. displacement: φ= F · X Const. magnetic induction: B · N = B · N on ∂B 0 Alg. 4 PD-PP Per. displacements: φ⁺= φ⁻ Per. magnetic potential: ϕ⁺ = ϕ⁻ on ∂B 0 Alg. 5 CT-LP Const. traction: P · N = P · N Lin. magnetic potential: ϕ = H · X on ∂B 0 Alg. 6 Reuss ^∗ Const. traction: P = P Lin. magnetic potential: ϕ = H · X in B 0

Fig. 3. Unit cell of a fiber composite and the associated finite element discretiza- tion. The volume fraction of the inclusion fiber is f = 25% .

in the unit-cell microctructure which undergoes 10% of simple- stretchinx-direction[F]xx=1.1andamagneticﬁeldinx-direction [H]x=50. It can be observed that the three different boundary conditionsresult invery similar homogenizedstresses and mag- neticinductions.Themaximumdifferencebetweenthemisbelow

2%. The microscopic answers are also similar, although the CT-CI boundaryconditionsresultinslightlylowermaximalstressandin- ductionvalues.

In the next example, the same unit cell is subject to 10% of simple-shear deformation inthe xy-plane [F]xy=0.1anda magnetic ﬁeld in the x-direction [H]x=50. The distributions of the microscopic shear stress, normalized by its macroscopic counter- part, and the microscopic magnetic induction in the x-direction, normalizedbythemacroscopicmagneticinduction,arepresented in Fig. 5. It is again found that the homogenized stresses and magneticinductionsareverysimilarforallthreeboundarycondi- tions.However, themacroscopicmagneticinductionsareapproxi- mately16%higherthanintheprevioussimple-extensionexample.

Thisis dueto themagneto-mechanical couplingeffectinsimple- extension.Thedistancesbetweentheparticlesincreaseandthere- fore the homogenized magnetic permeability decreases. The mi- croscopicdistributionsofthestressesandthemagneticinductions and their maximum values show small differences for the three different boundary condition. The highest microscopic stress and inductionvalues areobtained fortheLD-LP boundaryconditions.

Theinﬂuenceofthedifferentboundaryconditionsonthehomog-

Fig. 4. Macroscopic stretching [ F ] xx = 1 . 1 and magnetic ﬁeld [ H ] x = 50 : distribution of microscopic normalized Piola stress ( xx -component) and normalized magnetic induction in x -direction for applied LD-LP, PD-PP and CT-CI boundary conditions.

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Fig. 5. Macroscopic shearing [ F ] xy = 0 . 1 and magnetic ﬁeld [ H ] x = 50 : distribution of microscopic normalized Piola stress ( xy -component) and normalized magnetic induction in x -direction for applied LD-LP, PD-PP and CT-CI boundary conditions.

Fig. 6. Variations of the (a) macroscopic Piola stress ( xx -component) and macroscopic magnetic induction (in x -direction) due to the increase of the stretch for the simple- extension under magnetic ﬁeld [ H ] x = 50 and (b) macroscopic Piola stress ( xx -component) and magnetic induction (in x -direction) due to the increase of the magnetic ﬁeld under simple-extension [ F ] xx = 1 . 1 load, for applied LD-LP, PD-PP and CT-CI boundary conditions.

enizedstressandmagneticinductionisnot veryhighinthisunit cellexample,howeverthedifferencesinthemicroscopicﬁeldsand theirmaximumvaluesaremorepronounced.

Fig.6comparesthegraphsofthemacroscopicmacroscopicPi- olastress(xx-component)andmacroscopicmagneticinduction(in x-direction)obtainedfromtheincreaseofthestretch[F]xx forthe simple-extensionandmagneticﬁeld[H]x.Itcanbeobservedthat, also under increasing mechanical and magnetic loads, the three differentboundary conditionsresultin verysimilar homogenized stresses and magnetic inductions. The highest difference belongs to the variation of the macroscopic Piola stress versus the magnetic ﬁeld, where the CT-CI b.c. underestimates the results ob- tainedfromtheotherboundaryconditions,Fig.6b(left).

3.2. Periodic,randommono-disperseandpoly-disperse microstructuresundermagneticﬁeld,stretchingandshearing

Inthis sectionthe microscopicbehavior ofthreedifferent microstructures, periodic,random mono-disperse andrandom poly-

Fig. 7. Three different (a) periodic, (b) random mono-disperse and (c) random poly- disperse RVEs with the same volume fraction of f = 25% .

disperse, compare Fig. 7, is analyzed. The volume fraction of all microstructures is 25% andperiodic (PD-PP)boundary conditions areapplied.

Fig.8showstheobtainedmicroscopicPiolastressandthemag- netic induction in x-direction, normalized by their macroscopic counterparts, for the RVEs that undergo 10% of simple-extension

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Fig. 8. Macroscopic stretching [ F ] xx = 1 . 1 and magnetic ﬁeld [ H ] x = 50 : distribution of microscopic normalized Piola stress ( xx -component) and normalized magnetic induction in x -direction for different periodic, random mono-disperse and random poly-disperse microstructures.

deformationand amagnetic ﬁeld of[H]x=50.The homogenized values for the Piola stress and the magnetic induction are very similarforallthreemicrostructuresanddifferatmostby2%.How- ever, the microscopic distributions of stress and induction show large variations: the periodic microstructure represents a more uniform distribution of the microscopic quantities, whereas the randommono-disperseandrandompoly-disperseRVEsshowquite differentmicroscopicresponses.

Itcanbeobservedthattheproximityofparticlesalignedinthe directionoftheappliedmagneticﬁeldandalsotheexistence ofa smallparticleinthevicinityofabiggerone,leadtohighervalues ofthemicroscopicmagneticinductionandconsequentlytohigher microscopic stresses. The maximum stress values in the random RVEsaretwo timeshigherthanthe maximumstressinthe periodicmicrostructure.

Fig.9presentsthedistributionsofthemicroscopicshearstress andthemicroscopicmagneticinductioninxdirection,normalized bytheirmacroscopiccounterparts.TheRVEsareloadedby10% of simple-sheardeformationin thexy-plane, [F]xy=0.1anda mag- neticﬁeldinx-direction,[H]x=50.

The homogenized Piola stresses and magnetic inductions are againvery similar for all three microstructures. The macroscopic inductionisforallRVEshigherthaninthepreviousexamplewith simple-extensionalthoughthesamemagneticﬁeldisapplied.The reasonisthe sameasintheunit cellexample: Thedistancesbe- tween the particles are increasedin simple-extension andthere- fore the homogenized magnetic permeability decreases. The mi- croscopicdistributionsoftheshearstressandthemagneticinduc- tionsdiffersigniﬁcantly:themaximummagneticinductioninthe periodicmicrostructure ishalf ofthat in therandom microstructures.Thenormalizedmicroscopicshearstressintheperiodicmi- crostructurerangesfrom−4.90to5.17whereastherandommono- dispersemicrostructureshowsnormalizedshearstressesbetween

−12.37and11.83.Thedecisivefactorforthemagnitudeofthemi- croscopicinductionandconsequentlyalsoforthemagnitudeofthe stressisthedistancebetweentwoparticlesinthedirectionofthe magneticﬁeld.Therefore,highmagneticinductionandstress val-

uesareobtainedfortherandommicrostructuresatcertainpoints, whicharecanceledout duringhomogenizationbutarecrucialfor themicroscopicbehaviorofthematerial.

The curves of homogenized Piola stress and magnetic induction for the periodic, mono-disperse and poly-disperse microstructures obtained from PD-PP b.c. are compared in Fig. 10. The numerical results prove that the homogenized Piola stresses and magnetic inductions are very similar forthe three different microstructures.

It can be summarized,that the microscopic material behavior of amagneto-active compositestrongly dependson the distribution of the particles. The macroscopic material behavior, e.g. the homogenized stresses, strains or magnetic fields observed in an experiment, mightbe similar forvariousmicrostructures, butthe microscopic loading can be significantly different depending on theminimumdistancesofparticles.Consideringthemodelingand simulationofmagneto-activematerials,itseemstobeveryimpor- tant to representthemicrostructures asaccurately aspossible to computethecorrectmaximumvaluesofstressesandmagneticin- ductionsonthemicro-scale.Ifoneismainlyinterestedintheho- mogenized material answer a periodic unit cell gives sufficiently accurateresults.

3.3. Convergenceofhomogenizedvariablesfordifferentboundary conditions

Theobjectiveofthissectionistocomparethenumericalresults obtainedfromtheapplicationofdifferentboundaryconditionsfor increasingsizesoftheRVEs.Fig.11(a)and(b)illustratesmultiple sizesof periodic andrandom poly-disperseRVEs. Notethat their sizeincreasesinsuchawaythatthehigherlevelincludestheun- derlying lowerlevels. Theresponse oftherandom microstructure ofaparticularsizeisobtainedbysolvingandaveragingtendiffer- entpoly-disperse RVEs.Fig.11(c)showsan exampleof thoseten samplesforthesize1x1.Intheﬁrststepthesameboundarycon- ditions as in the previous unit cell example are applied, i.e. de- pendingontheprimary variablesofthemagneto-elasticenthalpy

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Fig. 9. Macroscopic shearing [ F ] xy = 0 . 1 and magnetic ﬁeld [ H ] x = 50 : distribution of microscopic normalized Piola stress ( xy -component) and normalized magnetic induction in x -direction for different periodic, random mono-disperse and random poly-disperse microstructures.

Fig. 10. Variations of the (a) macroscopic Piola stress ( xx -component) and macroscopic magnetic induction (in x -direction) due to the increase of the stretch for the simple- extension under magnetic ﬁeld [ H ] x = 50 and (b) macroscopic Piola stress ( xx -component) and magnetic induction (in x -direction) due to the increase of the magnetic ﬁeld under simple-extension [ F ] xx = 1 . 1 load, for periodic, mono-disperse and poly-disperse RVEs.

functionW(^F,H)^.^Secondly,^boundary^conditions^are^prescribedⁱⁿ termsoftheprimaryvariablesofthemagneto-elasticinternal en- ergyfunctionW^∗(^F,B)^.

3.3.1. Boundaryconditionsbasedonprimaryvariablesofenthalpy functionW(^F,H)

Fig.12illustratesthevariationofmacroscopicquantities,i.e.the enthalpyW,thePiolastress[P]_xxandthemagneticinduction[B]_x, versus thesizeofaperiodic RVE.The microstructuresare subject to twodifferentloadcases,one withsimple-extension[F]xx=1.1 andamagneticfield[H]_x=7.6inx-directionandthesecondone withnodeformation [F]=Ianda magneticfield [H]x=6.6 inx- direction.Thetop rowofFig. 12showsthenumericalresultsob- tainedforthefirstloadcase,thebottomrowthoseforthesecond loadcase.

In the top row of Fig. 12, it is ﬁrstly observed that the re- sultsofallboundaryconditionsconvergetotheresultsofthepe- riodicboundaryconditions,asexpected.TheLD-LPboundarycon- ditions renderhigher averagedquantities than the PD-PPbound-

aryconditions,andtheCT-CIboundary conditionsresultinlower averaged values. The results convergefor increasing sizes of the RVEsatdifferentratesforthevariousboundaryconditions.Forthe magneto-elasticenthalpyandthestress,theLD-LPboundarycon- ditionsconvergemuchfasterthanthe CT-CIboundaryconditions.

Incontrast,forthemagneticinduction,theCT-CIboundarycondi- tionsshowaslightlyfasterconvergence.TheVoigt’sapproximation givesthehighestresultsinall threecases,theReuss’ approximation thelowest one. The second load case, i.e.witha prescribed deformation gradient of[F]=I, is considered in thebottom row ofFig.12.Theresultsforall boundaryconditionsalsoconvergeto theresultsoftheperiodicboundaryconditionsforincreasingsizes oftheRVEs.However,somedifferencesintheorderoftheresults canbeobserved.Thehomogenizedmagneto-elasticenthalpy,com- putedwiththeLD-LPboundaryconditionsisalwayssmaller than the result of the PD-PP boundary conditions, and the magneto- elasticenthalpyderived withtheCT-CIboundaryconditionsisal- wayshigher.The Voigt’sandReuss’ approximationsshow asimi- larbehavior, i.e.Voigt’s approximationrendersasmallerenthalpy

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Fig. 11. Different sizes (levels) of (a) periodic and (b) random poly-disperse microstructures with the same volume fraction of f = 25% . The size increases in such a way that the higher level includes the underlying lower levels and the volume fraction remains constant.(c) shows ten random poly-disperse RVE of size 1x1.

Fig. 12. Comparison of the macroscopic enthalpy W , macroscopic magnetic induction [ B ] x and macroscopic Piola stress [ P ] ^xx , obtained from the application of LD-LP, PD-PP, CT-CI b.cs. and Voigt’s and Reuss’ approximation to the periodic microstructures. The size of the RVE increases from a unit-cell to a 10x10 periodic microstructure that contains all underlying lower size RVEs.

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Fig. 13. Comparison of the macroscopic enthalpy W , macroscopic magnetic induction [ B ] x and macroscopic Piola stress [ P ] xx , obtained from the application of LD-LP, PD-PP, CT-CI b.cs. and Voigt’s and Reuss’ bounds to the random microstructures. The size of the RVE increases from a 1x1 to a 10x10 random poly-disperse microstructure that contains all underlying lower size RVEs. Macroscopic responses are obtained from the solution and averaging ten different random poly-disperse microstructures. The bars in the diagrams indicate the standard deviation of the results of poly-disperse microstructures from the average solution.

andReuss’ approximationa higherone. The reasonfortheseun- usual results isfoundedin themagneto-elastic enthalpy function itself, which is poly-convex in F(when H=0) butconcave in H inthemagneto-mechanical case,compareFig. (2).Therefore,the Voigt approximation doesin general not resultin the maximum magneto-elastic enthalpy values. This is not observed in the top row ofFig. 12, since that exampleis dominated by the mechanical response.Consideringthe stress, theresultsfor boththe LD- LPandtheCT-CIboundaryconditionsapproachtheperiodicresult frombelow,wherebytheLD-LPconvergefaster.Thisisduetothe saddle-pointstructure ofthe enthalpy functiontogether withthe nonlinearityofthestressresponse.

Itisanalyzedinthenextexampleifthesamebehaviorcanbe observed forrandom poly-dispersemicrostructures.In Fig.13the homogenized magneto-elastic enthalpy, magnetic induction and stress are depicted for increasing sizes of random poly-disperse RVEs.Thesameloadcasesandboundaryconditionsasinthepre- viousexampleareanalyzed.

Forthepoly-disperseRVEstheperiodicboundaryconditionsdo not provide the exact solution and, therefore, the results of the periodic boundaryconditions alsochangewithan increasing size of the RVE. However, it can be observed in all cases, that they converge to a constant value (which is in general not the same as forthe periodic microstructure). The results of the LD-LP and theCT-CIboundaryconditionsalsoconvergeto theperiodicones.

The curves are not assmooth as in the example with the peri- odicmicrostructurewhichcomesfromtherandomnessofthepar- ticledistributions. Eachcurverepresentsthe averagevaluesof10 random poly-disperse RVEs of the same size. The error bars indicate thestandard deviation, whichdecreases forthelargerRVE sizes. Similar observationshave beenreportedby Bayat andGor- daninejad(2017).TheyobservedthatforsmallRVEsizestheran- dom positioning ofinclusionshas highinﬂuenceon theeffective responses.However,byincreasingtheRVEsizeandnumberofpar- ticlestheoscillationsanddeviationsoftheeffectiveresponsessig- niﬁcantlyreduce.Theorderoftheresultsforthedifferentbound- aryconditionsinFig.13is similarasforthe periodicmicrostruc-

tures.InthetoprowofFig.13themagneto-mechanicalloadcase is presented,where the mechanicalloading dominates the prob- lemandtherefore theusual behavior is observed,i.e.Voigt’s ap- proximationgivesthehighestvalues,followedbytheLD-LP,PD-PP andCT-CI boundary conditions,and Reuss’ approximation results inthe lowestvalues. Forthepoly-disperse RVEsthe convergence rates ofthe LD-LP andCT-CI boundary conditions are similar. In thebottomrowtheresultsformagneticloadingwithF=Iarede- picted.Again,thesaddle-pointstructureofthemagneto-elasticen- thalpybecomesapparent,sincetheReuss’approximationgivesthe highestmagneto-elasticenthalpyvalues,followedbytheCT-CI,PD- PPandLD-LPboundaryconditionsandVoigt’sapproximation.For themagneticinductionandthestress LD-LP boundaryconditions andVoigt’sapproximationoverestimatetheresponsewhiletheCT- CI boundary conditions and Reuss’ approximation underestimate them.Thebehaviorofthestressesisdifferentascomparedtothe periodicmicrostructure,sincetheLD-LPboundaryconditionsnow approach the periodic ones from above as expected. This is not onlyobservedfortheaveragevaluesbutalsoforallrandomRVEs and therefore the observation made for periodic microstructures seemstobeapeculiarityresultingfromtheregularmicrostructure.

Itcan be summarizedfortherandom poly-disperseRVEsthat the homogenized values obtained for different boundary conditions andvarious microstructures convergetocertain similar val- ueswhen thesize ofthe RVEis increased. The maximumdevia- tionsoccurintheRVElevelsbetween1× 1and6× 6andthedif- ferentcurvesstart toshow amore stableandsmootherbehavior forthe RVE sizesabove 7× 7 cells. Periodicboundary conditions showasexpectedthefastestconvergencetothehomogenizedval- ues, whereas the convergence rates of the LD-LP and the CT-CI boundaryconditionsaresimilar.

The observations for the magneto-elastic enthalpy and the stressesinthepurelymagneticloadcaseforperiodicandrandom microstructuresshowthatitisnotpossibleforthepresentsetting of the coupled magneto-mechanical problem to ensure that e.g.

Voigt’s approximation or the LD-LP boundary conditions always overestimatethehomogenizedquantitiesasitholdsforpurelyme-

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Fig. 14. Comparison of the macroscopic enthalpy W , macroscopic magnetic induction [ B ] x and macroscopic Piola stress [ P ] xx , obtained from the application of LD-CI, PD-PP, CT-LP b.cs. and Voigt ^∗and Reuss ^∗bounds to the periodic microstructures. The size of the RVE increases from a unit-cell to a 10x10 periodic microstructure that contains all underlying lower size RVEs.

chanical problems.Thismotivates theanalysis ofa second set of boundary conditions, formulated in the primary variables of the magneto-elasticinternal energyfunctionW^∗(^F,B) ^which^is ^poly- convexinFwhen(B=0)andconvexinB,inthenextsubsection.

3.3.2. Boundaryconditionsbasedonprimaryvariablesofenergy functionW^∗(^F,B)

Inthissubsection,boundaryconditionsformulatedinFandB, theprimary variablesofthemagneto-elasticinternalenergyfunc- tionW^∗areapplied,asintroducedinSection2.3.Thisismotivated bythefactthatthemagneto-elasticinternalenergyispoly-convex inFwhen(B=0) andconvexinB.Therefore,itisexpectedthat theVoigt^∗ andReuss^∗ approximationsdeliver boundson the ho- mogenizedbehavior.Thefollowingcombinationsofboundarycon- ditionsresult:lineardisplacements andconstantmagnetic induction(LD-CI),periodicdisplacementsandperiodicmagneticpoten- tial(PD-PP) andconstant tractions andlinear magnetic potential (CT-LP).ForVoigt^∗approximation,constant strainsandaconstant magneticinductionareassumedeverywhere intheRVE,whereas Reuss^∗ approximationprescribes constant stressesanda constant magnetic ﬁeld in the RVE. Fig. 14 presents the evolution of the averagedmagneto-elastic energyW^∗,theaveraged magneticﬁeld [H]x and the averagedPiola stress [P]xx versus the size of a periodic RVE. The composites undergo a magneto-mechanical load with[F]xx=1.1 and[B]x=0.01 (results in top row),anda mag- neticloadwith[F]=Iand[B]x=0.01(bottomrow).

TheupperplotsofFig.14showthattheresultsofallboundary conditions lie between Voigt^∗ and Reuss^∗ approximations where the LD-CI b.c. overestimate the results of the periodic boundary conditions and the CT-LP b.c. yield lower averaged values than theother two boundary conditions.Furthermore, all results converge to the (constant) results related to the periodic boundary conditions,whereby convergence isfaster forthe LD-CI b.c. than for the CT-LP boundary conditions. For the second load case in the bottom row, a similar behavior is observed forthe averaged magneto-elastic internal energy andthe averaged magnetic ﬁeld.

Sincethe magneto-elasticinternal energyW^∗ ispoly-convex inF (whenB=0)andconvexinB,Voigt^∗andReuss^∗assumptionsde-

liver asexpectedthehighestorlowest values,respectively.How- ever,fortheaveragedstressestheVoigt^∗assumptiondoesnotren- der the maximum value since the deformation gradient is prescribed as [F]=I in the whole domain of B0 which cancels out themechanicaltermofthePiolastressinEq.(9).Furthermore,the results ofboth the LD-CI andthe CT-LPboundary conditions ap- proachtheperiodiconesfrombelow.Thisbehaviorisagainacon- sequence ofthenonlinearityofthe coupledconstitutive relations anddependsonthemicrostructureandthematerialparameters.

Thehomogenizedmagneto-elasticinternal energyW^∗,thehomogenized magneticﬁeld [H]x and thehomogenized Piola stress [P]xx for poly-disperse RVEs of different sizes are considered in Fig.15.Theerrorbars indicateagainthe standarddeviations.The top rowshowsthe resultsforthemagneto-mechanical loadcase, thebottomrowtheresultsforthemagneticloadcasewith[F]=I. The three different boundary conditions are compared, comple- mentedbytheVoigt^∗ andReuss^∗approximations.Duetotheran- domnessofthepoly-dispersemicrostructure,thecurvesarenotas smoothasinFig.14fortheperiodiccase,butthevariationsofthe results andthe convergencebehavior are quite similar. Concern- ing the order of the results, all homogenized values,but the Pi- olastressinthesecondloadcase,showthehighestvaluesforthe Voigt^∗ approximation,followedbyLD-CI,PD-PP, CT-LPandReuss^∗ approximation.ForthePiolastress inthemagneticloadcase,the orderisdifferentandVoigt^∗approximationdoesnotgivethehigh- estvalue.

To summarize the convergence analysis forthe second set of boundaryconditions,formulatedinFandB,itcan bestatedthat asexpectedtheresultsofall boundaryconditionsconvergetothe same homogenizedquantities foran increasing size ofRVEs. The Voigt^∗ andReuss^∗ approximationsyield bounds for thehomoge- nizedenergyduetoitsconvexity,butnotforthehomogenizedPi- olastress.Thisisdueto thefactthatthe deformationgradientis prescribedas[F]=IinthewholedomainofB0 whichcancelsout themechanicaltermofthePiolastressinEq.(9)andthereforethe Voigt^∗ assumptiondoesnotrenderthemaximumvalueofthePi- olastress.

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Fig. 15. Comparison of the macroscopic enthalpy W , macroscopic magnetic induction [ B ] x and macroscopic Piola stress [ P ] xx , obtained from the application of LD-CI, PD-PP, CT-LP b.cs. and Voigt ^∗ and Reuss ^∗bounds to the random microstructures. The size of the RVE increases from a 1x1 to a 10x10 random poly-disperse microstructure that contains all underlying lower size RVEs. Macroscopic responses are obtained from the solution and averaging ten different random poly-disperse microstructures. The bars on the diagrams indicate the standard deviation of the results of poly-disperce microstructures from the average solution.

4. Conclusion

The behavior of heterogeneous magneto-rheological composites subjectedto large deformationsand external magneticﬁelds is studied. Computational homogenization is used to derive the macroscopic material response from the averaged response of theunderlyingmicrostructure.The microstructureconsistsoftwo materials and is far smaller than the characteristic length of the macroscopicproblem. Different typesof boundaryconditions based on the primary variables of the magneto-elastic enthalpy andinternalenergyfunctionalsareappliedtosolvetheproblemat the micro-scale.The overall responses of theRVEs withdifferent sizes and particle distributions are studied under different loads andmagneticﬁelds.

The ﬁniteelement resultsindicatethat theperiodic RVEsrep- resent a uniformmicroscopic response forthe applicationof different boundary conditionsandmagneto-mechanical loads. How- ever,themicroscopicﬁeldsobtainedfromrandommicrostructures showlargevariationsanddifferfromtheresultsofperiodicRVEs.

Thereby microscopic material behavior ofa magneto-active composite strongly depends on the distribution of the particles and canbe signiﬁcantlydifferentdependingonthedistances between themagneticparticles.Consideringthemodelingandsimulationof magneto-activematerials,itisveryimportanttorepresentthemi- crostructuresasaccuratelyaspossibletocomputethecorrectmax- imum values of stresses and magnetic inductions on the micro- scale.

Aconvergencestudyofthehomogenizedmacroscopicﬁeldsfor the boundary conditions based on the primary variables of the magneto-elasticenthalpyfunctionW(^F,H)^shows^the^dependency ofthemacroscopicresultsonthemagneto-mechanicalloadingand the microstructure of the RVE. For instance, under purely magnetic loadingVoigt’sapproximation rendersthesmallestenthalpy andReuss’approximation thehighestone.However, fortheRVEs undergoing magneto-mechanicalloads Voigt’s andReuss’ approximations result in the expected highest and lowest macroscopic magneto-elastic enthalpy, respectively. The reason for these un-

usual resultsis foundedinthe magneto-elasticenthalpy function itself, whichis poly-convex in F (when H=0) but concave in H inthemagneto-mechanicalcase.Thesamestudyiscarriedoutfor theboundaryconditionsbasedonthemagneto-elasticinternalen- ergyfunctionW^∗(^F,B)^.^The^results^representâ^consistent^response of the homogenized fields for different magneto-elastic loadings andmicrostructures.Sincethemagneto-elasticenergyW^∗(^F,B)îs poly-convexinF(whenB=0)andconvexinB,Voigt^∗andReuss^∗ assumptions always deliver the highest andthe lowest values of the homogenized energyas expected, respectively. However, this resultcannotbetransferredtothestresses,whichshowthehigh- estvaluesfortheperiodicboundarycondition.

Furthermore,increasingthesizesoftheRVEs,thehomogenized responses obtainedfromnon-periodic boundaryconditions based onW(^F,H)^and^W^∗(^F,B) ^converge^to^the ^results^associated^with periodicboundaryconditions.However,forthepoly-disperseran- dom microstructures the convergence behavior is not as smooth asthebehavior whentheparticlesareuniformlydistributed.Itis alsoobservedthat thelargerthesizeoftheRVE(biggerthan7x7 inourstudy),thesmoothertheoscillatoryresponseoftherandom microstructures.

Acknowledgments

The support of this work by the ERC Advanced Grant MO- COPOLYisgratefullyacknowledged.

AppendixA. Numericalimplementation

Inthe appendix the FE-formulation andnumericalimplemen- tationofthe homogenizationproblemare presented.Initially the FE-formulation for solving the magneto-mechanical homogenization problemis given andafterwards the algorithms forthe implementation of various types of mixedboundary conditions are discussed.