Single crystal anisotropy and coupled stability analysis for variant reorientation in Magnetic Shape Memory Alloys

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Krishnendu HALDAR, George CHATZIGEORGIOU, Dimitris LAGOUDAS - Single crystal

anisotropy and coupled stability analysis for variant reorientation in Magnetic Shape Memory

Alloys - European Journal of Mechanics A/Solids - Vol. 54, p.53-73 - 2015

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Science Arts & Métiers (SAM)

is an open access repository that collects the work of Arts et Métiers ParisTech

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To cite this version :

Krishnendu HALDAR, George CHATZIGEORGIOU, Dimitris C. LAGOUDAS - Single crystal

anisotropy and coupled stability analysis for variant reorientation in Magnetic Shape Memory

Alloys - European Journal of Mechanics A/Solids - Vol. 54, p.53-73 - 2015

Any correspondence concerning this service should be sent to the repository

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Single crystal anisotropy and coupled stability analysis for variant

reorientation in Magnetic Shape Memory Alloys

Krishnendu Haldar

, George Chatzigeorgiou

, Dimitris C. Lagoudas

b_{Department of Materials Science and Engineering, Texas A&M University, College Station, TX 77843, USA} c_{LEM3, UMR 7239, Arts et Mtiers ParisTech, Metz-Lorraine, 57078 Metz, France}

a b s t r a c t

Magnetic Shape Memory Alloys (MSMAs) have been the subject of much research in recent years as potential high-actuation-energy multifunctional materials. In this work we analyze coupled magneto-mechanical stability analysis of a variant reorientation mechanism for a single crystal based on a pro-posed 3-D magneto-mechanically coupled constitutive equations, derived in a consistent thermody-namic way. Discrete symmetry is considered to take into account single crystal anisotropy in the modeling. Analytical results are presented to demonstrate the influence of coupling and anisotropy in the stability of such a material system. Finally, a coupled Boundary Value Problem (BVP) usingfinite element analysis is performed by considering actual specimen geometry and boundary conditions that are used in the experiments. The numerical simulation reveals localization zones in the solutions due to the loss of ellipticity of the coupled magneto-mechanical problem.

1. Introduction

MSMAs are best known for their unique ability to produce Magnetic Field Induced Strains (MFIS) up to 10% under a magnetic

field (O'Handley, 1998; O'Handley et al., 2000; Müllner et al., 2003;

Shield, 2003). Some of the commonly used MSMA material systems

are NiMnGa (Murray et al., 2001a; O'Handley et al., 2003; Heczko

et al., 2003; Likhachev et al., 2004), FePd (James and Wuttig, 1998; Yamamoto et al., 2004) and NiMnX, where X ¼ In,Sn,Sb (Sutoua et al., 2004). The unique magneto-mechanical coupling makes MSMAs promising materials for multifunctional structures,

actuator and sensor applications (Pasquale, 2003; Tellinen et al.,

2002; Sarawate and Dapino, 2006, 2007; Karaman et al., 2007). The coupled MSMA behaviors can be modeled by considering the material as an electromagnetic continuum. Extensive work on different electromagnetic formulations had been proposed in the

literature (Toupin, 1956, 1960; Penfield and Haus, 1967; Hutter

et al., 2006; Eringen and Maugin, 1990) based on different notion of breaking up long range and short range forces. In a recent work,

(DeSimone, 1993; DeSimone and Podio-Guidugli, 1996) proposed a

continuum theory for deformable ferromagnetic materials.

Dorfmann and Ogden (2004, 2005)derived a theory of nonlinear

magneto-elasticity for magneto sensitive elastomers.McMeeking

and Landis (2005), McMeeking et al. (2007)presented a study of electrostatic forces on large deformations of polarizable materials. A theory for the equilibrium response of magneto-elastic

mem-branes is formulated bySteigmann (2004, 2009).Podio-Guidugli

et al. (2010)formulated a continuum theory for the evolution of magnetization and temperature in a rigid magnetic body for ferro/ paramagnetic transition. The variational formulations for general magneto-mechanical materials have been proposed by many

au-thors Kankanala and Triantafyllidis (2004), Ericksen (2006),

Bustamante et al. (2008), Miehe et al. (2011b,a).

The macroscopically observable MFIS in MSMAs is caused by the

microstructural reorientation of martensitic variants (O'Handley

et al., 2000; Karaca et al., 2006), field induced phase

trans-formation (Sutoua et al., 2004; Kainuma et al., 2006; Karaca et al.,

2007, 2009) or a combination of the two mechanisms. In this work we will focus on variant reorientation. In the variant reor-ientation mechanism, the variants have different preferred

di-rections of magnetization and the magneticfield is applied to select

certain variants among others, which results in the macroscopic shape change.

There are two major modeling approaches for variant reor-ientation mechanism. In microstructural based models, the resulting macroscopic strain and magnetization response are pre-dicted by minimizing a free energy functional. Details on the

microstructural based modeling approach can be found in (James

and Wuttig, 1998; DeSimone and James, 1997; O'Handley, 1998; Murray et al., 2001b; DeSimone and James, 2002, 2003). The sec-ond approach to study the material behavior is through

thermo-dynamics based phenomenological modeling (Hirsinger and

Lexcellent, 2003b, a; Kiefer and Lagoudas, 2005; Kiefer et al., 2006; Kiefer and Lagoudas, 2009). Most recent development of a variational modeling of variant reorientation in MSMAs can be

found inWang and Steinmann (2012). Modeling offield induced

phase transformation with detailed experimental characterization

can be found inHaldar et al. (2014). The effects of magnetic body

force and couple on the variant reorientation mechanism are

investigated through a coupled boundary value problem inHaldar

et al. (2011).

One of the major challenges for understanding the magneto-static response of the MSMAs is the experimental measurement of

the magneticfield inside the material. The measurements of the

magneticfield are strongly influenced by the shape and size of the

specimensShield (2003). In general, the shape of the specimen,

used in the experiments, is a prismatic bar with a rectangular cross section. A compressive stress is applied along the axis of the

pris-matic bar and a magneticfield is applied along the perpendicular

direction of the axisShield (2003), Kiefer et al. (2006). In order to

build a reliable constitutive model, one needs to relate the

measured data of the applied magnetic field with the magnetic

field in the material. A numerical technique to correct the consti-tutive responses from the demagnetization effect is also described inHaldar et al. (2011).

During the reorientation process, the nonuniformity caused by the shape effect combined with the strong nonlinear constitutive response in magnetization leads to localization of the numerical

solution. We investigated this phenomenon inHaldar et al. (2010),

where we found that the magnetic field during reorientation

changes drastically in the band like localized zones.

In this study, there are two major contributions. Thefirst is that

the considered constitutive equation includes single crystal anisotropy. A rate form for the mechanical and magnetization constitutive equations is derived to facilitate the stability analysis in

an efficient way. We analytically perform stability analysis for

magneto-mechanical coupling with anisotropy, where our previous

work was confined only within magnetostatic analysis with

isotropic assumption. We found that a material parameter due to

anisotropy indeed influences the stability of the system. The second

major point is the implementation of the coupled magneto-mechanical stability analysis. In the FE analysis, our main effort is to understand how instability caused by the magnetic behavior

affects the mechanicalfield variables. So, we only considered

isot-ropy in the FE analysis to solve a relatively simple problem.

Addi-tional efforts are necessary to see the in_{fluences of anisotropy in the}

numerical analysis, which are not considered in this study.

The organization of the paper is as follows: In Section 2, we

briefly introduce a thermodynamic based continuum framework

for a coupled magneto-mechanical dissipative system. A specific

form of the Gibbs free energy is proposed in Section 3 and the

constitutive equations are derived with discrete symmetry re-strictions. An incremental form of the constitutive equations with magnetic and mechanical tangent stiffness matrices is presented in 4 and the stability analysis of such a coupled system is presented in

Section 5. Some numerical examples are presented in Section6.

Finally, in Section 7 we solved a boundary value problem to

demonstrate the appearance of band-like structure due to loss of

ellipticity. Parameter identifications and validation of the newly

proposed constitutive equations are essential. As it is not our main focus, we present a concise but detailed descriptions in the B.

2. Continuum approach for single crystal MSMA modeling

In this section we briefly outline the thermodynamic framework

to describe the constitutive equations. Evolution equations of the internal state variables, introduced to capture the hysteretic behavior of the magneto-mechanical response, are also discussed.

2.1. Thermodynamic framework

We denote the reference configuration by

U

any externally applied stimuli and the current configuration by

U

The body consists of material points X₂

U

the deformed configuration is denoted by x ¼ 4(X,t) and the

deformation gradient is defined by F ¼ VX4 with J ¼ detF > 0. In the

deformed con_figuration

U

the magneticfield by h and the magnetization vector by m. The free

current (conductive) density of the body is neglected in this study. The constitutive response of MSMAs undergoing variant reor-ientation will depend on state variables such as appropriate

mea-sures of stress and magneticfield and also internal state variables to

account for loading path dependence due to the hysteretic response caused by dissipation. Using the Coleman and Noll procedure (Coleman and Noll, 1963), the following constitutive equations can be obtained E¼ r0G;SE; (1a) m₀M¼ r0G;H; (1b) s¼ G;T; (1c) rG;Zi$ _Zi⩾0: (1d) Here E¼1 2ðC  IÞ, C ¼ F T_F,

_r

0is the density, M¼ JF1m is the

magnetization, H¼ FT_{h is the magneticfield and fZg is the set of}

internal variables in

U

SE¼ JF1

_s

_s

_s

_s

_s

1960) generated due to combined magneto-mechanical effect (i.e. if

t is the total traction, then t¼

s

s

deformed configuration and T is the temperature. On deriving Eq.

(1), one needs to start from the internal energy and introduce a

proper Legendre transformation with regard to the deformation

gradient F and magnetic induction b in the deformed configuration.

Detailed derivations can be found in (Haldar and Lagoudas, 2014;

Haldar, 2012). Our next step is to define the set of internal

vari-ables_fZg.

2.1.1. Internal state variables

We assume that the volume fraction of the newly orientedfield

induced variant is

x

and the internal magnetization Mr, generated during variant

reor-ientation, are tensorial internal variables of order two and one respectively. The internal magnetization takes into account the phenomenological effect of different micro-magnetic mechanisms like the rotation of magnetization vector and the evolution of

magnetic domain walls. Finally, we consider the mixing energy grof

the reorientation as an internal variable. So the complete list of the

internal state variables isfZg ¼ fEr_{; M}r_{; x; g}r_{g. Expanding the}

p_Er: _Erþ p

Mr_{$ _}Mrþ px_x þ pgrgr_ 0: (2) The thermodynamic driving forces are denoted by

p_Er¼ rG ; Er; p_Mr¼ rG ; Mr; p_x¼ rG; x; pgr¼ rG; gr:

We further assume that the reorientation strain rates obey the

followingflow rules

_Er

¼ Lr_x: ₍₃₎

TheLr_{is the tensor which takes into account the direction and}

magnitude of the generated strain during variant reorientation. Similarly we consider that the rate of magnetization vector

generated during reorientation follows the followingflow rules

_

Mr¼ gr_{_x; (4)}

where

g

internal magnetization due to the evolution of

x

single scalar variable

x

inelastic reorientation tensor Erand 3 independent components of

the magnetization vector Mrthrough the evolution equations and

so the model is capable of taking into account any arbitrary magneto-mechanical loading conditions.

The evolution of interaction or mixing energy between the

martensitic variants during reorientation (_gr) can be represented

by

_gr_{¼ f}r_{_x; (5)}

where fris the hardening function.

The total thermodynamic driving force

p

reor-ientation is given by

pr_{¼ p}

Er: Lrþ p

Mr_$grþ pxþ pgrfr: (6)

The following transformation function,

F

F :¼

pr_{ Y}r_{; _x > 0}

pr_{ Y}r_{; _x < 0}; F  0; (7)

where Yris a positive scalar associated with the internal dissipation

during reorientation. The proposed reorientation function is similar to the transformation function used with conventional shape

memory behavior (Lagoudas et al., 1996; Qidwai and Lagoudas,

2000). It is assumed that the constraints of the reorientation

pro-cess follow the principle of maximum dissipation and can be

expressed in terms of the Kuhn Tucker type conditions (Simo and

Hughes, 1998)

F  0; F_x ¼ 0: (8)

3. Proposed Gibbs free energy and constitutive equations for variant reorientation

We denote the Gibbs free energy of the variant-1 and the

variant-2 by GV1_{and G}V2_{respectively. The Gibbs free energy of the}

transforming phase is denoted by GV1/V2_{. We write}

GSE; H; T; Er_{; M}r_{; x; g}_{¼ G}V1  SE; H; T þ GV1/V2  SE; H; T; Er_{; M}r_{; x; g}_; where GV1/V2  SE; H; T; Er_{; M}r_{; x; g}_{¼ x}h_GV2  SE; H; T GV1  SE; H; Ti þ GI_SE_{; H; E}r_{; M}r_{þ G}mix_ðgÞ:

GIand Gmix are the Gibbs free energies due to the

magneto-inelastic deformation and the energy due to the mixing of the two variants during reorientation. We determine the integrity basis of the scalar function G for the above mentioned tensorial argu-ments in the following subsections. Detailed group-theoretical development to obtain the integrity basis for such a single crystal

material system can be found inHaldar and Lagoudas (2014). Here

we briefly present the applied part of the theoretical development

in a aim to calibrate the new model for the stability analysis. 3.1. Finite symmetry restriction for single crystal

The most widely used material for variant reorientation mechanism

is Ni2MnGa. The martensitic phase has 10 M structure with I4/mmm

space group. The classical point group is 4/mmm (D4h). Thefive

mag-netic point groups are 4=mmm; 4=mmm; 4 =mmm; 4=mmm; 4=mmm.

Among them only the 4=mmm is ferromagnetic and the rest of the

members are antiferromagnetic (Cracknell, 1975). So we consider

4=mmm to generate the integrity basis.

There are three variants for tetragonal martensitic phase. We denote variant-3, which has shorter length (c) along the z direction.

The x and y axes are along the longer side a (Fig. 1(a)). The

ste-reographic representation of the group elements of 4/mmm is

shown inFig. 1(b). The notation of the diagram is followed from

Bradley and Cracknell (1972). Thefilled square (-) at the center represents the 4-fold rotations along the z axis, which are perpendicular to the plane of the paper and obey the right-hand rule. A solid ellipse ( ) denotes 2-fold rotation along the ii axis. The alpha-numeric labeling of the symmetry operations are placed

on thefigure in the position to which the letter E is taken by that

operation. C_4z± represents 360/4 anticlockwise/clockwise rotation

along the z axis and C2iis the 180rotation along the ii axis. We

select variant-1 (shorter axis is along the X1direction) by applying

traction on the single crystal along X1. The orientation of the initial

configuration of variant-1 is presented inFig. 2. The variant-2 has

the shorter length along the X2direction. When the magneticfield

intensity is high enough along the direction of spontaneous

magnetization (X2direction), variant-2 becomes preferred. We

as-sume that these two structural phases are magneto-elastic. 3.2. Energetics of variant 1

In this subsection, we discuss the coupled magneto-mechanical energetics of variant-1. The Gibbs free energy is considered as

GV1¼ GV1



SE; H; T0

and the elements of the integrity basis are

I1¼ H1; I2¼ H22þH23; I3¼ SE22þSE33; I4¼ SE11 I5¼

h

SE₃₁i2þhSE₁₂i2; I6¼hSE₂₃i2; I7¼ SE

At a given reference temperature T0, we consider the following quadratic expansion GV1ðI 1; I2; I3; I4; I5; I6; I7; I8Þ ¼ G10ðT0Þ  1 r0  a₁I₁þ a₂I2 1þ a3I2 þ a₄I2 3þ a5I24þ a6I5þ a7I6þ a8I7 þ a9I8þ a10I1I3þ a11I1I4 þ a12I3I4  ; (9)

and we write the mechanical and magnetic constitutive equations as1 E1¼ r0GV1; SE ¼ ð2a₅I₄þ a₁₂I₃þ a₁₁I₁Þi5i þ 2  2a₆SE 12þ a9H2  Sym½i5j þ2a₄I₃þ a₈SE 33þ a12I4þ a10I1  j5j þ 4a7SE 23Sym½j5k þ 22a6SE31þ a9H3  Sym½k5i þ  a8SE22þ a12I4þ a10I1 þ 2a4I3  k5k; (10) and m₀M1¼ r0GV1;H ¼a1þ 2a2H1þ a10  SE₂₂þ SE 33  þ a11SE11  iþ2a3H2 þ a9SE12  jþ2a3H3þ a9SE31  k: (11) 3.3. Energetics of variant-2

Similarly, considering the Gibbs free energy for variant-2 as

GV2¼ GV2



SE; H; T0;

the elements of the integrity basis are given by

J1¼ H2; J2¼ H32þH12; J3¼ SE33þSE11; J4¼ SE22; J5¼

h

SE₁₂i2þhSE₂₃i2; J6¼hSE₃₁i2; J7¼ SE

33SE11; J8¼ H3SE23þH1SE12

It should be noted that the global components of the elements of the integrity basis of variant-2 are different than variant-1 due to

different orientation. GV2_{can be expanded up to quadratic power as}

Fig. 1. (a) Orientation of the variant-3. x,y,z is the bodyfixed (local) coordinate system and X1,X2,X3is the global coordinate system (b) Stereographic representation of the symmetry

elements and local reference axes for 4/mmm point group.

Fig. 2. Orientations of variant-1 and variant-2.

1 _{The symmetric part of a second order tensor A is denoted by Sym}

½A ¼1 2ðA þ A

GV2ðJ1; J2; J3; J4; J5; J6; J7; J8Þ ¼ G2 0ðT0Þ _r01  b1J1þ b2J21þ b3J2 þ b₄J2 3þ b5J24þ b6J5þ b7J6þ b8J7 þ b9J8þ b10J1J3þ b11J1J4 þ b12J3J4  : (12)

The strain and magnetization can then be expressed as

E2¼ r0GV2;SE ¼2b4J3þ b8SE33þ b12J4þ b10J1  i5i þ 22b6SE12þ b9H1   Sym½i5j þ ð2b5J4þ b12J3þ b11J1Þj5j þ 2  2b6SE23þ b9H3   Sym½j5k þ 4b7SE₃₁Sym½k5i þ2b₄J₃þ b₈SE 11þ b12J4þ b10J1  k5k: (13) and m₀M2¼ r0GV2;H ¼2b3H1þ b9SE12  iþb1þ 2b2H2þ b10  SE 33þ SE11  þ b11SE22  jþ2b3H3þ b9SE23  k: (14)

3.4. Reorienting from variant-1 to variant 2

Considering the initial phase of the single crystal MSMA as the

stress favored variant-1 and applying magnetic_{field, field favored}

variant-2 nucleates. During the nucleation process, the two variants coexist and form twin structure. The internal strain and magneti-zation thus generated are taken into account by considering the

internal variables Erand Mr. We consider the Gibbs free energy of

this intermediate phase as

GI¼ GI_SE_{; H; E}r_{; M}r_{; T0}_:

Since variant-2 is nucleating, we impose symmetry restrictions

of variant-2 on the scalar function GI. Considering onlyfirst order

stress and inelastic strain coupling, the elements of the integrity basis can be written as

K₁¼ SE 33þ SE11; K2¼ SE22; K3¼ Er33þ Er11; K4¼ E22r K5¼ H2; K6¼ Mr2; K7¼ h SE₁₂i2þhSE₂₃i2; K8¼hSE₃₁i2 K9¼ SE33SE11; K10¼  E₁₂r 2þE₂₃r 2; K11¼E₃₁r 2 K12¼ Er33SE11; K13¼ SE31Er31; K14¼ h SE₃₃ SE 11 i E₃₃r  Er 11  K15¼ Er12SE12þ Er23SE23; K16¼ H23þ H21; K17¼  Mr₃2þMr₁2; K18¼ H3Mr3þ H1Mr1:

Moreover, consideringfirst order coupling between stress and

inelastic strain and betweenfield and internal magnetization, the

expanded form of the Gibbs free energy is written as

GIðK1; K2; K3; K4; K5; K6; K12; K13; K14; K15; K18Þ ¼ GI 0ðT0Þ  1 r0 ðc1K1K3þ c2K1K4þ c3K2K3þ c4K2K4þ c5K5K6 þ c₆K₁₂þ c7K13þ c8K14þ c9K15þ c10K18Þ: (15)

The constitutive equations for the above proposed scalar

func-tion GIare given by

EI¼ r0GI;_SE ¼c1  Er₃₃þ Er 11  þ c2E22r  c8  E₃₃r  Er 11  þ c6Er33  i5i þ 2c₉Er 12Sym½i5j þ  c₃Er 33þ E11r  þ c₄Er 22  j5j þ 2c9Er23Sym½j5k þ 2c7Er31Sym½k5i þ

 c1  E₃₃r þ Er 11  þ c₂Er 22þ c8  Er 33 E11r  k5k; (16) and m₀MI¼ r0GI;H¼ c5K6K5;Hþ c10K18;H ¼ c5Mr2jþ c10  Mr 1iþ M3rk  : (17) 3.4.1. Evolution equations

As the strain and magnetization evolution equations are con-cerned, we write the strain evolution equation as

_Er ij¼ Lrij

 S0E_x:

It can be shown that Lr

ij can be spanned as (Haldar and

Lagoudas, 2014) Lr ij¼ X p¼1 m dpðfIgÞDpij;

where, dp's are the scalar polynomials,

fIg ¼ fS0E 33þ S0E11; S0E22; ðS0E31Þ 2_{; ðS}0E 33 S0E11Þ 2_{; ðS}0E 12Þ 2_{þ ðS}0E 23Þ 2_{g and the}

elements of the setfD g are given by

D1_¼i 5i þ k5k; D2¼j5j; D3¼2S0E31Sym½i5k; D4_¼_S0E 33S0E11  ðk5k  i5iÞ; D5_¼2S0E 12Sym½i5jþ 2S0E23Sym½j5k:

Thus, spanningLr_{up to m¼ 5 in terms of the elements of fD g,}

we write Lr_{¼ d} 1ði5i þ k5kÞ þ d2j5j þ 2d3S0E31Sym½i5k þ d4  S0E₃₃ S0E 11  ðk5k  i5iÞ þ 2d5  S0E₁₂Sym½i5j þ S0E 23Sym½j5k  :

Similarly, for the magnetization evolution equation

_ Mr¼ gr 4  S0E_x; we can write gr i ¼ X p¼1 m d0_PðfIgÞDp i; wherefIg ¼ fS0E 33þ S0E11; ðS0E12Þ 2_{þ ðS}0E 23Þ 2_{g and}

D1_{¼ j; D}2_{¼ S}0E

12iþ S0E23k:

The expression for

g

gr_{¼ d}0 1jþ d02  S0E₁₂iþ S0E 23k  : 3.4.2. Mixing energy

We consider the Gibbs free energy for the mixing as

Gmixðgr_{Þ ¼ }1

rgr: (18)

The evolution of the reorientation hardening energy gris related

with a hardening function fr. The selection of such a function is

discussed in B.1.3 for a specific example.

3.5. Vectorization of matrices

The final combined forms of the strain and magnetization

constitutive equations are written as

E¼ r0G;_SE ¼ E1þ xðDEÞ þ EI; (19)

M¼ r0

m₀G;H¼ M1þ xðDMÞ þ MI; (20)

where

D

_D

At this point we consider vectorization of matrices [E] and [SE],

which represents the constitutive equations in a convenient form

that is suitable for stability analysis. Moreover, since [E] and [SE] are

symmetric matrices, we consider half-vectorization. In a general

way, half-vectorization of a n n symmetric matrix A is denoted by

vechðAÞ¼hA11;…;An1;A22;…;An2;…;Aðn1Þðn1Þ;Aðn1Þn;Ann iT

:

We write for simplicityvechðAÞ¼ðAÞ and so



E1¼ ½AsSEþ ½AhðHÞ; (21) 

E2¼ ½BsSEþ ½B_hðHÞ: (22)

Here [As]6 6, [Bs]6 6and [Ah]6 3, [Bh]6 3can be determined

from the constitutive responses (10) and (13). (,) is 6  1 for a

symmetric tensor and 3 1 for a vector. Thus we write (19) as

ðEÞ ¼E1  þ xE2  E1  þEI  ¼ ½As  SEþ x½T₁SEþ ½A_hðHÞ þ x½T₂ðHÞ þEI; (23) where ½T1 ¼ ½Bs  ½As; ½T2 ¼ ½Bh  ½Ah:

Similarly, we can write from (11) and (14)

 M1¼ ½Ams  SEþ ½A_mhðHÞ þ ðpÞ; (24)  M2  ¼ ½BmsSE  þ ½BmhðHÞ þ ðqÞ; (25)

where, the dimensions of the above matrices are [Ams]3  6,

[Bms]3 6, [Amh]3 3and [Bmh]3 3and from (20)

ðMÞ ¼M1þ xM2M1þMI ¼ ½AmsSEþ x½T1m  SEþ ½AmhðHÞ þ x½T2mðHÞ þ ðT3mÞx þMI; where ½T1m ¼ ½Bms  ½Ams; ½T2m ¼ ½Bmh  ½Amh; ðT3mÞ ¼ ðqÞ  ðpÞ:

4. Tangent stiffness and incremental magneto-mechanical constitutive equations

We now derive the tangent stiffness for the incremental magneto-mechanical constitutive equations. By taking the time

derivative of Eq.(23)and write

 _E¼ ½As  _SE þ _x½T₁SEþ x½T₁_SEþ ½A_h_Hþ _x½T₂ðHÞ þ x½T2  _Hþ _EI ¼ ½½As þ x½T2  _SE þ ½½A_h þ x½T2  _Hþ½T1  SE þ ½T2ðHÞ þ ðLÞ_x; or  _SE ¼ ½ℂ_Eþ ½ℙ_Hþ ðDÞ_x; (26)

where½ℂ ¼ ½½As þ x½T11,½ℙ ¼ ½½As þ x½T11½½Ah þ x½T2 and

ðDÞ ¼ ½½As þ x½T11ð½T1ðSEÞ þ ½T2ðHÞ þ ðLÞÞ. Similarly,  _ M¼ ½Ams  _SE þ _x½T_1mSEþ x½T_1m_SEþ ½A_mh_H þ _x½T2mðHÞ þ x½T2m  _Hþ _xðT3mÞ þ _M I ¼ ½½Ams þ x½T1m  _SE þ ½½A_mh þ x½T_2m_Hþ½T_1mSE þ ½T2mðHÞ þ ðT3mÞ þ ðgÞ  _x; or  _ M¼ ½ℂm_SEþ ½ℙm_Hþ ðDmÞ_x; (27)

where ½ℂm ¼ ½Ams þ x½T1m, ½ℙm ¼ ½Amh þ x½T2m and

ðDmÞ ¼ ð½T1mðSEÞ þ ½T2mðHÞ þ ðT3mÞ þ ðgÞÞ. Substituting the

rela-tion (26) in (27) we obtain  _ M  ¼ ½ℂm_SEþ ½ℙm_Hþ ðDmÞ_x ¼ ½ℂm½ℂ_Eþ ½ℙ_Hþ ðDÞ_xþ ½ℙm_Hþ ðDmÞ_x ¼ ½ℂm½ℂ_Eþ ð½ℂm½ℙ þ ½ℙmÞ_Hþ ð½ℂmðDÞ þ ðDmÞÞ_x ¼ℂ0 m _E  þℙ0 m _H  þD0 m_x: (28)

Now, from the consistency condition (8) we consider two cases.

We write _FSE; H; x¼ 0 0ðF;_SEÞ_$  _SE þ F;H$ _Hþ F;x_x ¼ 0 (29)

and replacing _SEfrom (26) we obtain

ðF;_SEÞ_$  ½ℂ_Eþ ðDÞ_xþ½ℙT_F; SEþ F;H  $ _Hþ F;x_x ¼ 0 0_x ¼ ðF;SEÞ$½ℂ _Eþ  ½ℙT_F; SEþ F;_H  $ _H ðF;_SEÞ_{$ðDÞ þ F;x} : (30)

Substituting back (30) in (26) we have

 _SE ¼ ½ℂ_Eþ ½ℙ_H  ðDÞðF;SEÞ$½ℂ  _Eþ½ℙT_F; SEþ F;H  $ _H ðF;_SEÞ$ðDÞ þ F;x ¼ " ½ℂ  ðDÞ5½ℂ T_F; SE ðF;_SEÞ_{$ðDÞ þ F;x} # $  _E þ 2 4½ℙ ðDÞ5  ½ℙT_F; SEþ F;H  ðF;_SEÞ_{$ðDÞ þ F;x} 3 5_{$ _}H ¼ ½L$_Eþ ½K$ _H (31)

Here½L is a 6  6 mechanical tangent stiffness matrix and ½K is

a 6 3 magnetic stiffness matrix. Moreover, substituting back the

expression of _x in (28) we get  _ M  ¼ℂ0m _E  þℙ0m _H  D0 m  ðF;SEÞ_{$½ℂ _}Eþ  ½ℙT_F; SEþ F;H  $ _H ðF;_SEÞ_{$ðDÞ þ F;x} ¼ "  ℂ0 m    D0 m  5½ℂTF;_SE ðF;_SEÞ_{$ðDÞ þ F;x} # $ðEÞ_ þ 2 4ℙ0 m    D0 m  5  ½ℙT_F; SEþ F;H  ðF;_SEÞ$ðDÞ þ F;x 3 5_{$ _}H ¼ ½L0_ $  _Eþ ½K0_ $ _H (32)

where½L0_{ is a 3  6 mechanical tangent stiffness matrix and ½K}0_{ is a}

3 3 magnetic stiffness matrix.

Case-II: _x¼ 0 and

F

In this case,½L ¼ ½ℂ, ½K ¼ ½ℙ, ½L0_{ ¼ ½ℂ}0

m and ½K0 ¼ ½ℙ0m.

4.1. Small strain approximation and 2-D reduction of the problem Since, the experiments are performed in small strains, we

reduce the model into an infinitesimal strain model such that E z ε,

SEz

s

_s

_m

0H5M, H z h and M z m. We further reduce the

problem in 2-D, where the stress components are sE

11, sE22,sE12,

strain components areε11,ε22,ε12, the magneticfield components

are H1,H2, and the magnetizations are M1, M2. The reduced form of

the strain constitutive equations in the matrix and vector notation thus can be written as below

0 B @ ε1 11 ε1 12 ε1 22 1 C A ¼ 2 42a05 00 2aa126 a12 0 2a4 3 5 0 B @ sE 11 sE 12 sE 22 1 C A þ 2 4a011 a09 a10 0 3 5 H1 H2 ; (33) 0 B @ ε2 11 ε2 12 ε2 22 1 C A ¼ 2 42b₀4 0_{0 2b}b12₆ b12 0 2b5 3 5 0 B @ sE 11 sE 12 sE 22 1 C A þ 2 4_b0₉ b₀10 0 b11 3 5 H1 H2 : (34)

The expressions of [As], [Ah] and [Bs], [Bh], as discussed in Eqs.

(21) and (22)respectively, can be found directly from the above two relations for the 2-D case. Moreover, with the above mentioned

assumptions, Eq.(16)can be reduced to

εI_¼_c1εr 11þ c2εr22þ c8εr11  i5i þc3εr 11þ c4εr22  j5j:

Enforcing the isochoric condition, tr (εI_{)¼ 0, we can write ε}r

22¼ εr 11  c1þc3þc8 c2þc4  and obtain 0 B @ ε1 11 ε1 12 ε1 22 1 C A ¼ εr 11t1 0 @ 10 1 1 A; (35)

where t1¼c1c4þcc24þcc84c2c3. Similarly, the matrix forms of the

magne-tization equations are

m₀ M11 M1₂ ! ¼  a11 0 a10 0 a9 0 0 B @ sE 11 sE 12 sE 22 1 C A þ  2a2 0 0 2a3  H1 H2 þ a1 0 ; (36) m₀ M21 M2₂ ! ¼  0 b9 0 b10 0 b11 0 B @ sE 11 sE 12 sE 22 1 C A þ  2b3 0 0 2b2  H1 H2 þ 0 b1 : (37)

The expressions of [Ams], [Amh], (p) and [Bms], [Bmh], (q) as

mentioned in Eq.(24) and (25)respectively, can be obtained from

the above two relations. The components form of the internal magnetization can be expressed as

M₁I M₂I ! ¼ c10M1r c5M2r ¼ c10d01s0E11 c5d01 ¼ q0 p0 : (38)

Finally, the 2-D reduction of (31) is written

2 6 4_s E 11 _sE 12 _sE 22 3 7 5 ¼ 2 4LL1121 LL1222 LL1323 L31 L32 L33 3 5 2 4_ε11_{_ε12} _ε22 3 5 þ 2 4KK1121 KK1222 K31 K32 3 5 _H1 _H₂  (39)

and Eq.(32)reads

 _M₁ _ M2  ¼  L0₁₁ L0₁₂ L0₁₃ L0₂₁ L0₂₂ L0₂₃ 2 4_ε11_{_ε12} _ε22 3 5 þK110 K120 K₂₁0 K₂₂0  _H₁ _H2  : (40)

It should be noted that if we consider plane stress problem, the out of plane strain needs to be calculated and under plane strain the out of plane stress needs to be calculated and also the material parameters have a different interpretation.

5. 2-D stability analysis of the magneto-mechanical coupled system

Before performing stability analysis, wefirst list the system of

equations below V$H þ V$M ¼ 0 (41a) V  H ¼ 0 (41b) V$sE_{þ m} 0ðV$HÞH ¼ 0 (41c) sE__{¼ ½Lð_εÞ þ ½K _H (41d)} _ M¼ ½L0ð_εÞ þ ½K0_{ _H (41e)} ε ¼1₂J þ JT _(41f) V  J ¼ 0: (41g)

We introduce a tensor potential

J

displacement. Introducing a potential

J

coupled system as a system of_{first order PDEs. It should be noted}

that we replaced B from the Gauss Law, i.e,V,B ¼ 0, by means of the

constitutive equation B¼

m

s

s

symmetric.

These are 14 equations (in 2-D) and H1,H2,M1,M2,sE11; sE12; sE22,

ε11,ε12,ε22,

J

J

J

J

(41f) ε11¼ J11; ε22¼ J22; ε12¼1 2ðJ12þ J21Þ; (42) and so ½ ε  ¼ 2 6 6 4 J11 1₂ðJ12þ J21Þ 1 2ðJ21þ J12Þ J22 3 7 7 5: (43)

Taking the time derivative we can further write

0 @_ε11_ε12 _ε22 1 A ¼ 2 6 6 6 4 1 0 0 0 0 0 1 2 1 2 0 1 0 0 3 7 7 7 5 0 B B @ _J11 _J22 _J12 _J21; 1 C C A or ð_εÞ ¼ ½E  _J: (44) Thus, (41d) becomes  _sE¼ ½L½E  _J þ ½K_H_{; (45)}

and (41e) becomes

_

M¼ ½L0½E  _J þ ½K0__{_H}_{: (46)}

Finally, from (45) with the help of (41c), (46) with the help of

(41a), (41b) and (41g), we eliminate

s

write the system of equations as

2 6 6 4 ½H1 ½S1  H0 1   S0 1  ½C1 ½0 ½0 C01  3 7 7 5
ðHÞ ðJÞ ;1 þ 2 6 6 4 ½H2 ½S2  H0 2   S0 2  ½C2 ½0 ½0 C02  3 7 7 5
ðHÞ ðJÞ ;2¼ 0; where ½H1 ¼ ½1 0 þ ½1 0½K0; ½S 1 ¼ ½1 0½L0½E ; ½H2 ¼ ½0 1 þ ½0 1½K0_{; ½S 2 ¼ ½0 1½L}0_{½E ;}  H01  ¼ m₀  H1 0 H2 0  þ  1 0 0 0 1 0  K  ;  S01  ¼  1 0 0 0 1 0  L  E  ;  H0 2  ¼ m0  0 H1 0 H2  þ  0 1 0 0 0 1  K  ;  S0 2  ¼  0 1 0 0 0 1  L  E  ;  C1  ¼ ½0 1;  C0 1  ¼  0 0 1 0 0 1 0 0  ;  C1  ¼ ½1 0;  C01  ¼  1 0 0 0 0 0 0 1  :

The detailed derivation can be found in A. We denote the above system of equations in the following compact form,

AQ;1þ BQ;2¼ 0; (47)

where

Q

J

J

J

J

elliptic, parabolic or hyperbolic when the eigenvalues of the following characteristic equation are complex, equal or real, respectively

detðB  aAÞ ¼ 0; (48)

with

a

6. Analytic results of stability analysis

.Our stability analysis will be based on the magnetization

re-sponses inFig. 3, where the newly developed model predictions are

compared with Kiefer-Lagoudas model (Kiefer and Lagoudas,

2005).2The most important material constants for the model are

identified as Msat_{, K, H}C_{, H}M2

s , HMf2, H M1

s , HMf1,E, ε

max_,

_s

1, n2, n3,

n4(Table B.7).3We consider Msat, HCandE to non-dimensionalize

the remaining material constants and material parameters. The

required material parameters are presented inTable 1. The

non-dimensionalized material constants and material parameters are

given in Tables 2 and 3, respectively. In terms of space

non-dimensionalization, we consider bx1 ¼ x1/w, bx2 ¼ x2/l, w/l ¼

g

where w is the width and l is the length of the specimen. Magnetic

field is non-dimensionalized as bH1¼MbHsat1and bH2¼bHMsat2. Moreover

we define non-dimensional quantities l ¼m0ðMsatÞ

2

E , br ¼Er and

bs_¼s

E, where r¼vFvx.

Three case studies for stability conditions will be discussed next. First (Case-I) we only consider the magnetostatic problem for isotropy. Next we consider the coupled system with isotropy

(Case-II) andfinally (Case-III) the anisotropic effect on the stability for the

coupled system is considered.

6.1. Case-I: magnetostatic stability condition

We start stability analysis with a simple magnetostatic case.

2 _{The values of the material constants can be found in (Haldar et al., 2011).} 3 _{A brief description of parameter identification procedure is presented in}

6.1.1. Before reorientation

Considering magnetostatic equations before reorientation, we

can write Eq.(47)in a non-dimensional way as

g 2 6 41 b v ba1 v bH2 0 1 3 7 5 bH1 b H₂ ;b1 þ  0 ð1 þ bÞ 1 0  bH₁ b H₂ ;b2 ¼ 0: (49)

The characteristic polynomial Eq.(48)is given by

a2_{ b v}ba1 v bH2

a þ ð1 þ bÞ ¼ 0 (50)

for which the discriminant is

bD :¼ ðb þ 2Þ2Hb22 4ð1 þ bÞ 1 bH2₂ : (51) Since bH2¼MbHsat2¼H_H2CMb C yand bM C y< 1, for H22 [0,HC], 1 bH 2 2> 0.

Thus the sign of bD will depend on the numerator only. The

numerator (Nr) further can be written as

Nr¼ 4ð1 þ bÞ " ðb þ 2Þ2 4ð1 þ bÞHb 2 2 1 # : (52)

At H2¼ bH2¼ 0, bD< 0 since for this particular problem

b

bD < 0. So the system is elliptic in the entire range [0,HC_].

6.1.2. During reorientation

When reorientation starts, from (47) we can write

g 2 6 41 b MC_xD bMl br 0 1 3 7 5 bH1 b H2 ;b1 þ 2 6 6 4 0 1 D b M2l br ! 1 0 3 7 7 5 bH_Hb1₂ ;b2 ¼ 0: (53)

The characteristic polynomial (48) is given by

a2_Mb C xD bMl br a þ 1  D b M2l br ! ¼ 0 (54)

and so the discriminant

bD :¼ 2 4 Mb C xD bMl br !2 þ 4 DMb 2 l br 3 5  4: (55)

The elliptic/hyperbolic nature of the system depends on the sign

of bD. When bD 0, the system is hyperbolic and when bD< 0, the

system is elliptic.Fig. 4shows that the system behaves hyperbolic

at the beginning but recovers ellipticity around bH2¼ 0.7 or at 0.84

[T].

6.2. Magneto-mechanical stability condition 6.2.1. Case-II: isotropic medium

As a next step, we consider a coupled magneto-mechanical stability analysis by assuming an isotropic medium. The system of equations can then be written as

g 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 1 Mb C xD bMl br 0 0 0 0 0 1 0 0 0 0 0 lD bMεmax brb 1 ba2 5 ba12ba5 2  4ba2 5 ba 2 12   ba12=ba5 4ba2 5 ba 2 12  0 0 l bH2D bM b2_ba 5 0 0 0 1 ba₅ba₆ 1 ba₅ba₆ 0 0 0 0 1 0 0 0 0 1 0 0 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 0 B B B B B B @ b H1 b H2 J11 J12 J21 J22 1 C C C C C C A ;b1 þ 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 0 1 DMb 2 l br 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 ba5ba6 1 ba5ba6 l bH2D bM b2_ba 5 lD bMεmax brb 1 ba2 5 ba12ba5  ba12=ba5 4ba2 5 ba 2 12   2 4ba2 5 ba 2 12  0 0 0 0 1 0 0 0 0 0 0 0 0 1 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 0 B B B B B B @ b H1 b H2 J11 J12 J21 J22 1 C C C C C C A ;b2 ¼ 0; (56)

for which the characteristic equation is " a2_Mb C xD bMl br a þ 1  D b M2l br !#h a2_{þ 1}i2_{¼ 0: (57)}

For the real root, we only need to consider

a2_Mb C xD bMl br a þ 1  D b M2l br ! ¼ 0 (58)

and so the conditions of stability are the same as for the magnetostatic (Case-I). It should be noted that the nonlinear magnetization constitutive equation causes the unstable behavior to the entire magneto-mechanical system.

6.2.2. Case-III: anisotropic medium

This part is the main application of an anisotropic mechanical coupling. Since, we do not have anisotropic

magneto-mechanical experimental characterization for the model calibra-tion, we only assume the following (from (33), (34)) non zero material constants for simplicity:

0 B @ ε1 11 ε1 12 ε1 22 1 C A ¼ 2 42a05 00 2aa126 a12 0 2a4 3 5 0 B @ sE 11 sE 12 sE 22 1 C A (59) 0 B @ ε2 11 ε2 12 ε2 22 1 C A ¼ 2 42b₀4 0_{0 2b}b12₆ b12 0 2b5 3 5 0 B @ sE 11 sE 12 sE 22 1 C A þ 2 40₀ b₀10 0 0 3 5 H1 H2 : (60)

Recall that superscripts 1 and 2 are for the 1 and

variant-2 respectively. The coefficient b10couples H2withε211, i.e, a

piezo-magnetic coefficient for variant-2. Since both the variants have the

same tetragonal structure, one needs to satisfy that the Gibbs free

energy GV1¼ GV2_{. We can thus immediately write b4}¼ a_{5, b12}¼ a_12,

b5¼ a4and b6¼ a6for purely mechanical conditions (from (9) and

(12)).

The matrix form of the magnetization equations are (from (36), (37))

Fig. 3. Model prediction comparison of magnetization responses with Kiefer-Lagoudas model (red dotted, with demagnetization correctionHaldar et al. (2011)) and present model (blue continuous). A2B1is the y-component and A1B2is the x-component. (For

interpretation of the references to colour in thisfigure legend, the reader is referred to the web version of this article.)

Table 1

Material parameters (isotropic medium). MC y¼ m0ðKMsatÞHsM2, MCx¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðMsat_Þ2_{ ðM}C yÞ2 q

2a3¼m0b,b¼m0KMsat, b1¼m0Msat,DM¼ Msat MC

a1¼ m0 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðMsat_Þ2_{ b}2_H2 2 q ,cH22 [0,HC), and a1¼ m0MxC,cH22½HC; HMf2 a5¼ a4¼2E1, a12¼nE, a6¼1þn2E, b10¼ d4¼ 0, ~d ¼ εmax Table 2

Non-dimensionalized material constants. b HM2 s ¼ 1, bH M2 f ¼ HM2 f HC, bH M1 s ¼H M1 s HC, bH M1 f ¼ HM1 f HC Table 3

Non-dimensionalized material parameters (isotropic medium). b MCy¼ MC y Msat¼ m0KHC, bM C x¼M C x Msat¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ð bMCyÞ2 r 2ba3¼2am03¼ b, bb1¼ b1 m0Msat¼ 1, D bM¼ DM Msat ba1¼m0aM1sat¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 bH22 q ,cH22 [0, HC), and ba1¼m0aM1sat¼ bM C x,cH22 [HC, HfM2], p¼ 0 ba5¼ ba4¼12,ba12¼ n, ba6¼1þn2, ~d¼ εmax l ¼m0ðMsatÞ2 E ,br ¼rE,bs¼s  E

M1 1 M1₂ ! ¼ ~a₁ 0 ; (61) and M2₁ M2 2 ! ¼  0 0 0 b10=m0 0 0 0 B @ sE 11 sE 12 sE 22 1 C A þ 0 b1=m0 : (62)

The additional material constants for the anisotropic medium is

summarized in Table 4 and the non-dimensionalization is

pre-sented in Table 5. Moreover, like isotropy, we consider br ¼r

E1, l ¼m0ðMsatÞ2

E1 and bs

_¼s

E1. The non-dimensionalized system of PDEs

can then be written as

where, bℂ ¼ 1 4ba₄ba₅ ba2₁₂ bA32¼lD b M br ba12 ba₄εmaxþ 2  εmax_{þ bH} 2bb10   2bb10x bB12¼ 1 bℂba4 bb2 10b2x2 l  D b M2l br þ b bs _bb 10 br þ bbb10bℂ br  ba12εmax_{x þ 2ba} 4  εmax_{þ bH} 2bb10 ! bB₄₂¼ DMlb br 2ba5 ba4ε max_þba12 ba4  εmax_{þ bH} 2bb10  þba12 ba4 bb₁₀x

and the characteristic polynomial is in the following form

t6a6þ t5a5þ t4a4þ t3a3þ t2a2þ t1a þ t0¼ 0;

where the coefficients are

t6¼ 1t5¼  b MC_xD bMl br t4 ¼ 1þ2ba6 ba₄ þ ba₁₂ ba₄  bH2bb10x  DMb br D bMl csbb 10b t3 ¼  bH2bb10x  D b M bMC_xl br 2ba6 ba4 þba12 ba4 t2 ¼ba5 ba₄þ bH2bb10x ba₁₂ 2ba₄þ ba₆þ ba12 ba₄ 2 41 þ DMb2l br þ b c sbb 10 br 3 5t1 ¼ ba5 b MC_xD bMl ba4br Hb2bb10ba12 2ba4 xt0 ¼ 0 @ba5 ba₄ bb2 10b2 lba₄ 1 A  ba5D bM 2 l ba₄br  bs _ba 5D bM bb10b ba₄br 1 2D bM bb10bε max_{x  D}M bbH2bb 2 10b ba₄br :

Due to complex dependency of the polynomial coefficients on

the material parameters, unlike previous two cases, it will be

g 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 1 Mb C xD bMl br 0 0 0 0 0 1 0 0 0 0 0 bA32 2  ba₁₂ ba4 0 0 l bH2D bM b2bℂba 4 0 0 0 1 bℂba4ba6 1 bCba4ba6 0 0 0 0 1 0 0 0 0 1 0 0 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 0 B B B B B B @ b H1 b H2 J11 J12 J21 J22 1 C C C C C C A ;b1 þ 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 0 bB12 2ba4bb10xbℂb2 l ba12bb10xbℂb2 l 0 0 1 0 0 0 0 0 0 0 0 0 1 bℂba4ba₆ 1 bℂba4ba₆ l bH2D bM b2_bℂba4 bB42  ba12 ba4 2ba5 ba4 0 0 0 0 1 0 0 0 0 0 0 0 0 1 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 0 B B B B B B @ b H1 b H2 J11 J12 J21 J22 1 C C C C C C A ;b2 ¼ 0; (63) Table 4

Material constants (anisotropy). a5¼_2E1₁, a4¼2E12, a12¼ n12 E1 ¼ n21 E2, a6¼ 1þn12 E1 , b10 Table 5

Material parameters (anisotropy). ba5¼12,ba4¼2EE12,ba12¼ n12¼ n21E1 E2 ,ba6¼ 1 þ n12, bb10¼ Msat_b 10 b br ¼r E1, l¼ m0ðMsatÞ2 E1 ,bs _¼s E1

practically a tedious task to obtain the discriminant of a sextic equation. So, we like to analyze such a situation by numerically investigating the eigenvalues. If all the six eigenvalues are complex numbers, then the system will be elliptic, otherwise, with at least one real eigenvalue the loss of ellipticity will take place. We

perform a parametric study with the piezomagnetic coefficient of

variant-2, b10, to investigate the influence of anisotropic coupling

on the stability of the system. The result is shown inFig. 5. We

introduce a binary indicator

i

the eigenvalues are complex numbers,

i

system becomes elliptic. If at least one eigenvalue becomes real, it takes green color and loss of ellipticity occurs. As an example, in

Fig. 5(a), for bb10¼ 1.0  103, the material system becomes stable

only after bH_{z0:71, where for bb}10> 3.0  103, the system remains

unconditionally unstable. Note that the lower limit bH¼ 0:5 and the

upper limit bH¼ 0:75 are the non-dimensionalized values of

reor-ientation startfield (HM2

s ) and reorientationfinish field (HMf 2). The

entire scenario changes considerably by changing some material

parameters in the magnetic constitutive equations (Fig. 3).Fig. 5(a)

explains the effect of shifting the reorientationfinish field (HM2

f ) as

twice as the existing value. This means that in the latter case a less stiff constitutive equation is used, for which the material behaves more stable in the entire range of reorientation.

7. Finite element analysis of the

magneto-mechanically-coupledfield equations for MSMA

In the previous section we presented stability results for the non dimensionalized magneto-mechanical problem for MSMAs. All the analytical results were derived at a material point, based on the newly proposed constitutive responses. The results of the study show that the material instability may occur at a certain magnetic field or at a certain range of magnetic fields. The unstable behavior means that if a bulk material body is concerned and if there are singularities in the body, the disturbances due to the singularities propagate along the characteristics. This phenomenon is expected to be observed by solving an appropriate boundary value problem.

We now consider a specific problem with real material parameters

(Table 1) in the FE implementation. The value of the material

constants (as presented inTable B.7) are taken from (Haldar et al.,

2011).

Field equations, constitutive relations and boundary conditions

are summarized inTable 6. Note that the magnetization

constitu-tive equations depend only on the magnetic field and so do not

account explicitly the influence of stress variation on the

consti-tutive responses in the coupled problem. Moreover, the maximum

strainεmax_{is also assumed constant and so it is not coupled with}

the non-uniform stress distribution in the material.

The numerical analysis presented here was performed using the

COMSOL Multi-physics finite element software package. The

ge-ometry and boundary conditions of the considered model problem

are illustrated inFig. 6. This particular arrangement is motivated by

the experimental set up, explained inAppendix B. The

computa-tional domain may be regarded as the gap between the pole pieces

of an electromagnet of dimensions 26 mm 26 mm  26 mm for

which a uniform magneticfield of up to 2 T can be applied. Typical

specimen dimensions are 8 mm 4 mm  4 mm, or aspect ratios of

2:1:1, where the long axis is the x-direction. All the_{field equations}

are summarized inTable 6and more details can be found inHaldar

et al. (2011). A spatially constant magnetic potential

Fm

x ¼ Fmy ¼ 0; Fmz ¼ m0Hayx; (64)

is applied on all sides of the boundary. The mechanical boundary

conditions of the problem are illustrated inFig. 7, where txand ty

denote the mechanical traction on the boundaries along the x- and the y-directions, respectively. The compressive traction along the x-direction is imposed by constraining the vertical displacement U of

thev

U

v

U

finite element analysis. 7.1. Numerical results

Magneticfield (H2) distribution at an appliedfield value m0H2a¼

1:1 [T], for which the magnetostatic system becomes unstable, is

presented inFig. 8. It is numerically observed that two band-like

zones are created inside the specimen. An enlarged view of the

specimen is given inFig. 9(a) with the plots of volume fraction

x

the newly orientedfield induced variant (variant-2). In the figure,

m

specimen in a constitutive sense (see,Fig. 3) corresponding to the

appliedfield m0H2a¼ 1:1 [T] due to the shape and size effect of the

specimen geometry. We observe that at the top-left and bottom-right corners, the volume fraction almost reaches 1 while in the intermediate region, the volume fraction varies from 0 to 0.3.

Fig. 5. Effect of the piezomagnetic coefficient b10on stability during reorientation. The green region is unstable where the yellow region is elliptic and stable. (a) Magnetization

response as described inFig. 3is considered and (b) where the reorientationfinished field (HM2

f ) is taken as twice as of case-(a). (For interpretation of the references to colour in this

Moreover there are sharp changes in the values of

x

behavior in Fig. 9(b). The directions of magnetization vectors

change very sharply in the regions A1B1C1D1 and

A2B2C2D2. The appearance of the band like zones (Fig. 9)

during reorientation is due to loss of ellipticity. The discriminant D

that dictates the loss of ellipticity is plotted at

m

and is presented in Fig. 10(a). The Figure shows that for

m

and loss of ellipticity occurs. The stable elliptic regions (‘E’ in

Fig. 10(a)) with D< 0, which are separated by the unstable hyper-bolic regions, have a completely different behavior in terms of the

field variables, like the magnetic field H2(Fig. 10(b)), the martensitic

variant volume fraction (Fig. 9(a)) and the magnetization vector

(Fig. 9(b)).

The values of the characteristic angles in the unstable regions in

the non-dimensional spatial description are given by the Eq.(57)

and they vary spatially. In the present study the two

character-istic angles of all the critical points are almost the same (60

and64_{in the actual specimen dimensions). The magneticfield}

shows a drastic change across characteristics that start from the top

right and bottom left corners (Fig. 10(b)).

In the solution of strain (onlyε11component is presented here)

(Fig. 11(a)), we observe the banded regions. Similar trend is also

observed in the stressfield4_{(Fig. 11(b)). We like to emphasize here}

that the magnetization constitutive equation is assumed to depend only at a constant stress level. This means the variation of stress does not have any impact on the magnetic constitutive response. The main magneto-mechanical coupling in the system is through

Table 6

Summary of thefield equations, constitutive equations and boundary conditions. Maxwell equations:

DFm_{¼ m} 0V  M.

Conservation of Linear and Angular Momentum: V$sE_m 0H(V$M) ¼ 0, skw(sE)¼ 0 . Constitutive Equations: M2¼ M2(H2), M1¼ M1(H2) sE_{¼ C : ðε  ε}r_Þ2Sym withε ¼1 2ðVu þ VuTÞ , εr¼ Lrxand Lr_{¼ ε}max 2 410 1 00 0 0 0 0 3 5: Boundary Conditions: EBF$n ¼ 0; EHF  n ¼ 0 ; Es þ sM_{F$n ¼ 0 where} sM_{¼ H5B }m0 2ðH$HÞI 0sE_{n¼ t}a_þm0 2ðM$nÞ2nþ m0ðH5MÞn :

Fig. 6. Domain geometry, mesh and boundary conditions for the magnetostatic problem.

Fig. 7. Imposed mechanical boundary conditions.Uis the material domain andvUits boundary.

Fig. 8. Magneticfield distribution (H2) inside and outside the specimen at the applied

field m0Ha2¼ 1:1 [T].

4 _{We plotted stress values only in the range 0e7 MPa. Level curves for higher}

Fig. 9. (a) Volume fraction offield induced variant and (b) magnetic field H2atm0<H2> ¼ 0.67 [T].

the Maxwell equations, conservation of linear and angular momentum.

Finally, the system becomes elliptic everywhere at the end of reorientation and so becomes stable. The banded zones thus

disappear from the solutions (Fig. 12).

8. Discussion

The coupled magneto-mechanical stability analysis shows that

magnetic instabilities influences the mechanical field variables. The

magnetic field singularities that occur at the corners of the

specimen and propagate along the characteristics, intensifies the

mechanical field variables in the banded regions. We observe

around7 MPa stress in the banded zones (Fig. 11(b)) which is even

higher than the blocking stress (5 MPa). Moving a little bit across

those defect lines, the magnitude of the mechanicalfield variables

suddenly jumps to a different value. However, as the stress

dependence in the reorientation strain, i.e., εmax_{(see Table 6) is}

assumed constant in the current FE analysis, the blocking stress effect on strain could not be captured and we observe considerable amount reorientation strain at the highly stressed regions (Fig. 11(a)).

Fig. 11. Distribution of (a)ε11and (b)s11atm0<H2> ¼ 0.67 [T](hyperbolic).

Depending on situations, these unstable phenomena can have possible positive or negative impacts in the devices level appli-cations. For example, in the current experimental conditions, a 2

[T] appliedfield may be considered as high. One way to reduce

this value is to provide some boundary defects, such as small V-notches, which will facilitate to nucleate and propagate more field singularities inside the materials and most of the interior

regions will befilled up with the unstable zones. Such a

phe-nomenon then increases the interior magnetic field intensities

and facilitate variant reorientation at a low appliedfield. Note

that singularities also increase the stress in the unstable zones and tries to suppress variant reorientation if the magnitude be-comes more than the blocking stress. So, to obtain the best desired effects, one needs to optimize among the geometry with

surface defects and the maximum/minimum field-stress

requirements.

On the other hand, such an effect for Field Induced Phase

Transfer (FIPT) (Haldar et al., 2014), where the blocking stress is

much higher (z130 MPa) than variant reoriantation, may lead to

a failure mechanism. The underlying mechanism in FIPT is phase transformation from anti-ferromagnetic martensite to the

ferro-magnetic autenite by applying ferro-magneticfield. Unfortunately, the

appliedfield required for complete phase transformation is very

high (_{z15 [T]). If we consider magnetic instability for such a}

material system, the stress in the banded regions could be large enough to cause a failure of the material. However, stability analysis of FIPT is not performed yet and we like to investigate the related issues in a future work.

9. Conclusions

We propose a magneto-mechanical coupled constitutive response for variant reorientation in single crystal MSMA. Discrete symmetry is taken into account to implement single crystal anisotropy. The developed constitutive equations are then considered in the coupled stability analysis. The analytic approach of stability analysis shows that the system becomes unstable during martensitic variant reorientation due to highly nonlinear magnetization constitutive response. We also have found that the single crystal anisotropy is an important factor and magneto-mechanical coupling due to anisotropy strongly

influences the stability conditions. An analytic parametric study

reveals that a material parameter which appears in the consti-tutive equations due to anisotropy could be responsible to make such a magneto-mechanically coupled material system unstable. FE analysis is performed to solve a coupled BVP. However, we only consider isotropy in the FE analysis as we aim to see the

influence of magnetic instability on the mechanical field

vari-ables through coupling. The results show that the appearance of

banded zones in the spacial distribution of the magnetic _field

variables as well as in the mechanicalfield variables, when loss

of ellipticity occurs. Acknowledgments

The authors would like to acknowledge thefinancial support of

the Army Research Office, Grant no. W911NF-06-1-0319 for the

initial stages of this work, NSF-IIMEC (International Institute for Multifunctional Materials for Energy Conversion) under Grant No. DMR-0844082, and NSF-NIRT, Grant no. CMMI: 0709283 for the

support of thefirst author.

Appendix

Appendix A. Calculations for the system of 1st order PDEs Eq.(41a)can be written as

½ 1 0  H1 H2 ;1 þ ½ 0 1  H1 H2 ;2 þ ½ 1 0  M1 M2 ;1 þ ½ 0 1  _M 1 M2 ;2 ¼ 0; or ½1 0ðHÞ_;1þ ½0 1ðHÞ_;2þ ½1 0½L0_{½E ðJÞ} ;1þ ½1 0½K0ðHÞ;1 þ ½0 1½L0_{½E ðJÞ} ;2þ ½0 1½K0ðHÞ;2¼ 0; or ½H1ðHÞ;1þ ½S 1ðJÞ;1þ ½H2ðHÞ;2þ ½S 2ðJÞ;2¼ 0: (A-1)

Similarly, (41c) can be written as

_{1 0 0} 0 1 0 0_B B @ sE 11 sE 12 sE 22 1 C C A ;1 þ _{0 1 0} 0 0 1 0_B B @ sE 11 sE 12 sE 22 1 C C A ;2 þ m0 _H 1 0 H2 0  _H 1 H2 ;1 þ m0 _{0 H} 1 0 H2  _H 1 H2 ;2¼ 0; or _{1 0 0} 0 1 0  sE ;1þ _{0 1 0} 0 0 1  sE ;2þ m0 _H 1 0 H2 0  ð H Þ_;1 þ m_{0 0 H} 1 0 H2  ð H Þ_;2¼ 0; or  1 0 0 0 1 0  ½L½E ðJÞ_;1þ  1 0 0 0 1 0  ½KðHÞ_;1þ  0 1 0 0 0 1  ½L½E ðJÞ_;2 þ _{0 1 0} 0 0 1  ½KðHÞ_;2þ m0 _H 1 0 H₂ 0  ðHÞ_;1þ m0 _{0 H} 1 0 H₂  ðHÞ_;2¼ 0; or we simply write  H0 1  ðHÞ_;1þS01  ðJÞ;1þH02  ðHÞ_;2þS02  ðJÞ;2¼ 0: (A-2) From (41b) we get H2;1 H1;2¼ 0; and so ½ 0 1  H₁ H2 ;1þ ½ 1 0  H₁ H2 ;2¼ 0; or ½C1ðHÞ;1þ ½C2ðHÞ;2¼ 0: (A-3)

Finally, Eq.(41g),V  J ¼ eijkJmj;iek5em¼ 0, gives two

J11;2 J12;1¼ 0; (A-4) J21;2 J22;1¼ 0: (A-5) So we can write  0 0 1 0 0 1 0 0 0_B B @ J11 J22 J12 J21 1 C C A ;1 þ  1 0 0 0 0 0 0 1 0_B B @ J11 J22 J12 J21 1 C C A ;2 ¼ 0; or  C01  ðJÞ;1þ  C02  ðJÞ;2¼ 0 (A-6)

Combining (A-1), (A-2), (A-3) and (A-6) we write

2 6 6 4 ½H1 ½S1  H01   S01  ½C1 ½0 ½0 C0 1  3 7 7 5
ðHÞ ðJÞ ;1þ 2 6 6 4 ½H2 ½S2  H02   S02  ½C2 ½0 ½0 C0 2  3 7 7 5
ðHÞ ðJÞ ;2¼ 0:

Appendix B. MSMA experiments and magneto-mechanical responses

In a typical experiment, the MSMA sample, initially in austenitic phase, is subjected to a constant compressive mechanical load along

the x-axis to transform into stress induced variant (Fig. B.13). Then a

magnetic field is applied along the perpendicular y-axis. After a

critical value, field induced variant nucleates and both variants

coexist by forming a twinned microstructure. Complete variant

reorientation tofield induced variant takes place at a high applied

magneticfield.

Fig. B.13. Schamatic of the experimental setup. Traction is applied in the vertical di-rection and magneticfield is applied in the horizontal direction.

As explained inFig. B.14, we consider(A) an ideal stress favored

variant with single magnetic domain in the initial configuration.

180domain walls in the variant-1 is assumed to be eliminated by

applying a smallfield along the easy axis.

When magneticfield is applied along the y-direction(hard axis),

the magnetization vectors start rotating. Once the criticalfield for

the variant reorientation has been reached, the_{field favored variant}

nucleates. In this configuration, 90 _{domain forms due to the}

presence of twin variants (O'Handley et al., 2003; Tickle, 2000).

After complete reorientation, only _{field induced martensitic}

variant is present and the magnetization process becomes satu-rated. The magnetization vectors are aligned along the applied

magneticfield, which is the easy axis of the field favored variant.

Fig. B.15. (a) Experimental data of strain-field response at 1.4 MPa. The four critical magnetic fields for forward and reverse orientation and maximum inelastic strain are shown. (b) Magnetization response of stress favored martensitic variant at a high stress level (3 MPa), such that no reorientation takes placeHeczko (2005).

Under these circumstances, the x-component of the applied

magneticfield is assumed to be zero. The magnetic field along the x

direction due to the magnetization of the body is assumed to be

small and the dependence ofM on H1is neglected. The

magneti-zation components are assumed to have the form of M1¼ M1(H2)

and M2¼ M2(H2). So, the components of thefield variables are in

the following form

H ¼ f0; H2g; M ¼ fM1; M2g: (B-1)

We also assume that the only non-zero stress component issE

11,

which is uniform and constant inside the specimen during the experiment.

Appendix B.1. Material parameter identi_fication

We consider that the variant-1 is under axial traction along the x

direction and magnetic field is applied along the y direction, i.e,

sE_{¼ s}E

11i5i and H ¼ H2j. Calibrations from different

magneto-mechanical parameters are given below.

Appendix B.1.1. Mechanical response

Due to lack of complete experimental data for the full aniso-tropic calibration of the model with the present magneto-mechanical loading conditions, we only consider the x-compo-nent of the elastic and inelastic responses. No stress dependence on

Lr_{is considered, i.e,L}r_{¼ d}

1i5i þ d2j5j þ d1k5k and so εr¼Lr

x

Then Eqs.(33)e(35)become

ε1_{¼ 2a5s}E 11i5i; ε2¼  2b4sE 11þ b10H2  i5i; εI_{¼ ~dxði} 5i  j5jÞ:

The total number of unknown parameters are a5,b4, b10,

~d ¼ εr

11t1¼ d1t1. We further assume a5 ¼ b4 ¼21E, No

magneto-elastic coupling, i.e, b10 ¼ 0. We set t1d1 ¼ εmax, which can

be found fromFig. B.15(a). Thus, the strain response can be reduced

to ε11¼ ðε1Þ11þ xðε2 ε1Þ11þ  εI 11¼ 1 EsE11þ εmaxx: (B-2)

Appendix B.1.2. Magnetic response

The expression of the magnetization can be written as

m₀ M1 M2 ¼ m0 M 1 1 M1 2 ! þ xm0 " M2₁ M2 2 !  M11 M1 2 !# þ m0 M I 1 MI 2 ! : (B-3)

We assume no stress dependence in the magnetization of

variant-1 and variant-2, i.e, a11¼ b10¼ 0 and Variant-2 always be in

saturation state, i.e, b2¼ 0. Then Eqs.(36)e(38)become

m₀M1¼ a1iþ 2a3H2j; ; m0M2¼ b1j; m0MI¼ pxj; (B-4)

and parameters to be identified are a1, a3, b1and p¼ c5d01.

Fig. B.16. Schematic of magnetization response.

A typical experimental observation of the magnetization

com-ponents is shown in Fig. B.16. MC

y is the y-component of the

magnetization at the beginning of reorientation, i.e, at HC_{¼ H}M2

s .

Knowing the experimental magnetization response (y-component)

of a pure stress favored martensitic variant, as shown inFig. B.15(b),

we can immediately identify saturation magnetization Msatand the

slope of the magnetization response K. Since OA (Fig. B.16) is the

linear response of the stress favored variant, MC

ycan be calculated

from HM2

s and the slope K as

M_yC¼ m0 

KMsatHM2

s : (B-5)

We can write directly, from the y-component of (B-4)

2a3¼ m₀MCy

HC ¼ m0b: (B-6)

Knowing the parameters of M2(y-component) from the

exper-iments, we will now determine the parameters pertaining to M1

response if the micro-scale mechanism followsFig. 14. Since the

magnetization vectors are only rotating in the region 0< H2 HC

(Fig. 14, Case-B), we always need to satisfyM1 ¼ Msat, and so we

can write

 M₁12þM₁22¼Msat2; 0a2 1þ 4a23H22¼  m₀Msat2_; (B-7)

and then with the help of (B-6),

a1¼ m0 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðMsat_Þ2_{ b}2_H2 2 q ¼ m0~a1: (B-8) Note that MC

x, the x-component of the magnetization at the

beginning of reorientation (H2¼ HC), can be written as

M_xC¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðMsat_Þ2__MC y 2 r : (B-9)

Since variant-2 is always saturated, from (B-4), b1 ¼

m

which also implies p0¼ 0 as

m

m

x

magnetization response takes the following form:

M1₁ M1 2 ! ¼ 8 > > > > < > > > > :  0 0 0 b  H1 H2 þ ~a₁ðH2Þ 0 ; for H2 HC_; MC_x MC_y ! ; for x2ð0; 1Þ: (B-10a) M2₁ M2₂ ! ¼ 0 Msat ; (B-10b) Mr 1 M₂r ¼ 0 p0 x: (B-10c)

Appendix B.1.3. Thermodynamic driving force

The four critical magneticfields are: the start of forward

reor-ientation, HM2

s , the end of forward reorientation, HM_f2, the start of

reverse reorientation, HM1

s , and the end of reverse reorientation,

HM1

f (Fig. B.15a). The reduced form of the thermodynamic force (6)

is given by

pr_{¼ s}E

11εcurþ m0DM2H2þ fr rDu0; (B-11)

where _rDu0¼ rG20 rG10. The hardening function is chosen as

Lagoudas et al. (2011) fr_:¼ 8 > > < > > : A 2  1þ xn1 ð1  xÞn2 B; _x > 0; C 2  1þ xn3 ð1  xÞn4_{ D; _x < 0;} (B-12)

where, n1,n2,n3and n4are some given exponents.

We need to know the parameters A,B,C,D,Yrand

rD

Kuhn Tucker condition (8) we obtain two conditions at the

begin-ning and two conditions at thefinish of the forward reorientation.

They are pr_s_{; H}M2 s   Yr_{¼ 0; for _x > 0; at x ¼ 0 (B-13a)} pr_s_{; H}M2 f   Yr_{¼ 0; for _x > 0; at x ¼ 1 (B-13b)}

Similarly, for reverse reorientation we get two more equations,

pr_s_{; H}M1 s  þ Yr_{¼ 0; for _x < 0; at x ¼ 1 (B-14a)} pr_s_{; H}M1 f  þ Yr_{¼ 0; for _x < 0; at x ¼ 0 (B-14b)}

The constant stress level is denoted by

s

hardening functionLagoudas et al. (1996)gives us

Z1 0 fr_{_x > 0}dx¼ Z1 0 fr_{_x < 0}dx: (B-15)

Solving the abovefive equations (from B-13a to B-15), we get the

solution of_{five unknowns, A; ~B, C; ~D; Y}r_{. It should be noted that we}

introduce a new constant ~B¼ B þ rDu0 and ~D¼ D þ rDu0since B

and D absorbs the term

rD

in theAppendix C. All the material constants are summarized in

Table B.7. Table B.7 Material constants. m0HMs2¼ 0:5 [T], m0HMf2¼ 0:58 [T], m0HMs1¼ 0:28 [T], m0HMf1¼ 0:1 [T] Msat_{¼ 742 [kA/m], K ¼ 1.25 [T]}1_{, H}C_{¼ H}M2 s ,εmax¼ 5%, E ¼ 2.0 [GPa] n1¼ 6.6, n2¼ 1.4, n3¼ 0.8, n4¼ 0.3,s*¼ 1.4 [MPa] m0¼ 1.2566  106[N/A2]