journal of magnetism and magnetic materials · 2020. 1. 22. · 38 p.g. evans, m.j. dapino /...

Journal of Magnetism and Magnetic Materials 330 (2013) 37–48

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Journal of Magnetism and Magnetic Materials

0304-88

http://d

n Corr

E-m

journal homepage: www.elsevier.com/locate/jmmm

Measurement and modeling of magnetic hysteresis under field and stressapplication in iron–gallium alloys

Phillip G. Evans, Marcelo J. Dapino n

Department of Mechanical & Aerospace Engineering, The Ohio State University, Columbus, OH 43210, USA

a r t i c l e i n f o

Article history:

Received 1 March 2012

Received in revised form

22 September 2012Available online 12 October 2012

Keywords:

Galfenol

Hysteresis

Thermodynamic model

Nonlinear characterization

53/$ - see front matter & 2012 Elsevier B.V. A

x.doi.org/10.1016/j.jmmm.2012.10.002

esponding author. Tel.: þ1 614 688 3689; fax

ail address: [email protected] (M.J. Dapino).

a b s t r a c t

Measurements are performed to characterize the hysteresis in magnetomechanical coupling of

iron–gallium (Galfenol) alloys. Magnetization and strain of production and research grade Galfenol are

measured under applied stress at constant field, applied field at constant stress, and alternately applied

field and stress. A high degree of reversibility in the magnetomechanical coupling is demonstrated by

comparing a series of applied field at constant stress measurements with a single applied stress at

constant field measurement. Accommodation is not evident and magnetic hysteresis for applied field and

stress is shown to be coupled. A thermodynamic model is formulated for 3-D magnetization and strain. It

employs a stress, field, and direction dependent hysteron that has an instantaneous loss mechanism,

similar to Coulomb-friction or Preisach-type models. Stochastic homogenization is utilized to account for

the smoothing effect that material inhomogeneities have on bulk processes.

& 2012 Elsevier B.V. All rights reserved.

1. Introduction

Iron–gallium alloys (Galfenol) with 12–29at % gallium are anemerging class of magnetostrictive materials which have both highmagnetostriction (� 400 ppm) and ductility similar to steel [1].This unique combination of mechanical robustness and highcoupling makes Galfenol ideal for creating sensors and actuatorsoperable in tension, bending and torsion in harsh environments.Furthermore, Galfenol can be machined, welded, extruded, anddeposited into innovative geometries.

Previous studies of Galfenol magnetomechanical behaviorinclude linear characterization [2,3] as well as a study of thenonlinear anhysteretic response [4] and reversibility [5]; a non-linear description including hysteresis has not been published. Inthis work, magnetization and strain measurements of productionand research grade Galfenol from Etrema Products, Inc. arepresented. Measurements include applied magnetic field at con-stant stress, applied stress at constant magnetic field, and alter-nately applied magnetic field and stress. Attention is given tokinematic reversibility, referring to dependence on stress andmagnetic field application order, and to thermodynamic irrever-sibility or the energy loss associated with magnetic hysteresisunder both applied stress and magnetic field.

A previous study of reversibility by Yoo et al. [5] presentsmeasurements with magnetic flux density and stress as

ll rights reserved.

: þ1 614 292 3163.

independent quantities and calculates the resulting energy trans-duction with magnetic flux density and strain as the independentquantities. Unaccounted-for energy was attributed to "anoma-lous" transduction, possibly associated with the choice of inde-pendent variables. Here the independent quantities are stress andmagnetic field.

The measurements presented here show a remarkable degreeof kinematic reversibility in the magnetomechanical coupling,even in the production grade sample. This is in contrast with themagnetomechanical coupling in steel which has been shown toexhibit stress and field-induced magnetization that is boththermodynamically and kinematically irreversible [6,7]. The kine-matic reversibility in Galfenol is demonstrated by comparing asingle stress-induced magnetization curve at constant magneticfield with a series of field-induced magnetization curves atconstant stress. Minor loop measurements consisting of decreas-ing the field from a bias point, decreasing stress from a bias point,returning the field, and returning the stress do not exhibit anyobservable accommodation. These measurements indicate thatmagnetic hysteresis for both applied field at constant stress andapplied stress at constant field results from the same physicalmechanism.

A new modeling framework is needed for magnetostrictivematerials like Galfenol which respond in a nonlinear, anisotropic,and hysteretic manner to applied field and stress yet are kinema-tically reversible. While many models have been developed forferromagnetic hysteresis and magnetomechanical coupling, a vec-tor hysteresis model for anisotropic, magnetostrictive materialsapplicable for both field and stress loading is not available.

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Strain gage

Drive coil

Pick−up coil

Hall probe

Push−rod

Flux return

Bushing

Load framewith load cell

Specimen

Fig. 1. Experimental setup for measuring magnetization and strain of Galfenol

under magnetic field and stress.

P.G. Evans, M.J. Dapino / Journal of Magnetism and Magnetic Materials 330 (2013) 37–4838

Common approaches for modeling hysteretic materials includedifferential equation-based approaches derived from energy prin-ciples as in the Jiles–Atherton model [8,9] and empiricalapproaches like the Preisach model [10]. Scalar in its classicalform, the Jiles–Atherton model has been generalized as a vector,anisotropic magnetization model as well as a scalar magnetome-chanical model [11,12]. The magnetomechanical model applies tokinematically irreversible behavior which has been reported fornickel, steel, and silicon iron [6,7]. The Preisach model is also scalarin its classical form and has likewise been generalized for isotropicvector magnetization [13–15], anisotropic vector magnetization[16] and scalar stress-induced magnetization [17,18]. The Preisachmodel has also been used to calculate magnetostriction [19,20].Recent work has combined aspects of the differential equationsapproach and the Preisach approach resulting in a model thatbenefits from the simplicity of the former and the generality of thelatter [21]. An alternative to these classical approaches is an energyweighting approach for giant magnetostrictive materials compris-ing an anhysteretic vector model for both field and stress applica-tion with a scalar hysteresis mechanism [22–25]. The hysteresismechanism uses concepts from the Jiles–Atherton model andhence inherits minor loop properties which do not agree withmeasurements.

Preisach-like models based on energy principles have beendeveloped by replacing the simple Preisach hysteron which takesvalues of 71 with multi-state relays with states calculated fromenergy principles. In a class of vector models the states representStoner–Wohlfarth particle orientations which have anisotropicdependence on magnetic field [26,27]. A homogenized energyframework applicable to ferroic materials utilizes a hysteronderived from the balance of exchange and thermal energies ofmagnetic moments and incorporates scalar magnetic hysteresisfor both field and stress application [28,29]. This framework hasbeen extended in an anisotropic 2-D vector implementation forelectrostrictive materials [30]. In general, energy-based modelswhich utilize a Preisach-like switching mechanism provide physicalinsight and exhibit hysteresis properties which agree withmeasurements.

In this work, a formal thermodynamic development is under-taken to construct a hysteron which is applicable to magnetos-trictive materials of arbitrary anisotropy. The hysteron dependson the 3-D magnetic field and stress and includes a small numberof parameters, each with a clear physical interpretation. Thenumber of hysteron states is dictated by material symmetry andanisotropy with one state for each magnetically easy axis. Thecriterion for switching follows from the second-law of thermo-dynamics and results in a unified hysteresis model which has thesame properties as observed in the measurements. Following theapproach of Smith et al. [28,29], a statistically distributed inter-action field is superimposed on the applied field. Rather thanemploy a 3-D, statistically distributed coercive field, a scalarcoercive energy is used in the homogenization scheme resultingin a reduction of the integration order from six to four ascompared to 3-D Preisach models. This model enables accuratedescription of the measurements and provides a frameworkfor understanding hysteresis in ferromagnetic materials whichexhibit kinematically reversible magnetomechanical coupling.

2. Measurements

Measurements include magnetization and strain of /100Soriented, production and research grade Galfenol samples fromEtrema Products, Inc. in cylindrical rod form with dimensions of0:25� 1 inches. In the measurement system, compressive stress isapplied with an MTS 858 tabletop system by loading the sample

between parallel plates. Magnetic field is applied with a drive coilsituated with the Galfenol sample in a steel canister providing aflux return path. Since the permeability of Galfenol is similar tothat of the steel return path, it is necessary to use a feedbackcontroller for the magnetic field [31,32]. The changing permeabilityof Galfenol results in a nonlinear relationship between the voltageapplied to the drive coil and the magnetic field in the Galfenolsample. Drive coil voltage is supplied by a Test Star II MTScontroller. The level of voltage required to achieve a desiredreference field is found by measuring the field with a Lakeshore421 gauss meter and implementing a PI controller with an NI SCB-69 DAQ acquisition board. Magnetic flux density is measured witha Walker Scientific MF-30 fluxmeter and pick-up coil. Magnetiza-tion is calculated by subtracting the magnetic field measurementfrom the flux density measurement. An illustration of the measure-ment setup is shown in Fig. 1.

2.1. Production Grade 18.4 at% Ga

Major magnetization loops for applied magnetic field at con-stant stress and applied stress at constant field are shown inFig. 2. Magnetization versus field measurements have two regionsof noticeable hysteresis, one at low fields and another at higherfields. Consider for example the curve obtained under 79 MPacompressive stress. There is a hysteretic region below 1 kA/m andanother between 7 and 12 kA/m. The higher field region moveshigher with increasing stress. The magnetization versus stress, atlow stresses, depends strongly on the field history. The curves inFig. 3(a) are measured after applying a positive saturating fieldand subsequently lowering the field to the bias point while thecurves in Fig. 3(b) are measured after applying a negativesaturating field and subsequently bringing the field to the biaspoint. In the latter, multiple loops overlap while in the former, thefirst loop does not close. An exception is the case where there iszero bias field after having negatively saturated the material. Inthis case, the negative remanence resulting from bringing thefield from negative saturation to zero is wiped out by a singlecycle of stress. Further cycles of the stress do not result in anymagnetization change. Reported measurements of steel have avery different response, exhibiting significant accommodationwith each stress cycle and a gradual convergence to alimiting loop.

Measurements shown in Fig. 4 show a remarkable degree ofreversibility in the magnetomechanical coupling of Galfenol. In4(a) a magnetization versus stress curve at constant field isobtained from direct measurement and compared to magnetiza-tion versus stress points obtained from multiple sets of magne-tization versus field measurements at different bias stresses. The

−60 −50 −40 −30 −20 −100

200

400

600

800

1000

Stress, MPa

Mag

netiz

atio

n, k

A/m

IncreasingField

−60 −50 −40 −30 −20 −10

0

200

400

600

800

1000

Stress, MPa

Mag

netiz

atio

n, k

A/m

IncreasingField

Fig. 3. Magnetization measurements of production grade Fe81.6Ga18.4 at constant field (0, 1.6, 2.4, 3.3, 4.0, 4.8, and 5.6 kA/m). (a) Bias field reached from positive

saturation; (b) bias field reached from negative saturation.

−20 −10 0 10 20

−1000

−500

0

500

1000

Magnetic Field, kA/m

Mag

netiz

atio

n, k

A/m Increasing

Stress

−20 −10 0 10 20−1000

−800

−600

−400

−200

0


Stra

in×1

06

IncreasingStress

Fig. 2. (a) Magnetization and (b) strain measurements of production grade Fe81.6Ga18.4 at constant compressive stress (0, 11, 19, 26, 35, 44, 52, 61, 70 and 79 MPa).

P.G. Evans, M.J. Dapino / Journal of Magnetism and Magnetic Materials 330 (2013) 37–48 39

points are obtained from the upper and lower branches of themagnetization versus field hysteresis curves at the given biasfield. These points are then plotted versus the respective biasstresses at which the magnetization versus field curves aremeasured. The overlap of the curve measured directly whileapplying stress and the points obtained from magnetizationversus field curves at constant stress suggests that the magneto-mechanical coupling is reversible and that magnetic hysteresisfrom applied field and applied stress results from the samephysical mechanism. It also suggests that the anomalous energyfound in [5] does not arise from Galfenol. Accordingly, thethermodynamic framework developed in Section 3 includes onlyenergy transduction between the magnetic and mechanical statesalong with energy losses arising from domain reconfiguration.

Fig. 4(b) shows measurements from alternately applied field andstress along with two magnetization versus stress curves at constantfield which are biased by first positively saturating the material andthen lowering the field to the bias point. A stress is then cycled threetimes to a 70 MPa compression. As before, the first loop is notclosed. The curve from alternate stress and field application isinitialized in the same manner by saturating positive and loweringthe field to 2.8 kA/m. Thus, as stress is applied it follows themagnetization versus stress curve with the higher bias field. At12.6 MPa compression the stress is held constant and the field islowered to 2 kA/m from its previous value of 2.8 kA/m. This movesthe magnetization from the upper branch of the magnetizationversus stress curve at the higher bias field to the upper branch of themagnetization versus stress curve at the lower bias. Here the field isagain held constant and the stress is relaxed to 9.4 MPa resulting ina shift of the magnetization towards the lower branch of the majormagnetization versus stress loop. The stress is again held constantand the field is returned to 2.8 kA/m bringing the magnetization tothe lower branch of the magnetization versus stress curve obtained

with a 2.8 kA/m bias field. The stress is then returned to 12.6 MPa atconstant field, moving the magnetization towards the upper branchof the magnetization versus stress major loop. This stress and fieldcycle is repeated three times before returning the stress to zero.These minor loops obtained by alternately applying field and stressabout a bias point lack any noticeable accommodation. To empha-size, this is in contrast with measurements of steel which exhibitlarge changes in the shape of magnetization loops obtained fromcycling the stress and field [6,7].

2.2. Research grade 18.5 at% Ga

Figs. 5–7 show measurements for the research grade material.Major magnetization loops of this material have similarities withthe production grade material. Specifically, the field location ofthe burst region is moved to higher fields with increasing biasstress (see Fig. 5(a).) The loops, however, do not have a noticeablehysteretic region at low fields as do the production grade magne-tization loops. Additionally, the differential susceptibility is higher inthe hysteretic burst regions. The magnetization versus stress loops(see Fig. 6(a)) are similar to the production grade measurementswhere the first cycle is not closed and subsequent cycles overlap.They differ in the high slope regions, having much sharper transi-tions. The magnetomechanical coupling in the research gradematerial is shown to be reversible by comparing the magnetizationversus stress measured at constant field and the magnetizationversus field measured at constant stress in the same manner as wasdone with the production grade material (see Fig. 7(a).)

The coupled nature of the hysteresis mechanism observed inthe magnetization from both applied field and applied stress isemphasized in Fig. 7(b). Starting from a field of 4.2 kA/m and acompressive stress of 15 MPa, if the magnetization starts at pointA on the upper branch of the major magnetization versus field


loop, increasing the stress and returning to 15 MPa pushes themagnetization to point B on the lower branch of the major loop ina single cycle. Further cycles always return to point B on the lowerbranch and overlap. Starting at point B on the lower branch of the

−50 −40 −30 −20 −10 00

200

400

600

800

1000

Stress, MPa

Mag

netiz

atio

n, k

A/m

Constant Field

Constant Stress

−70 −60 −50 −40 −30 −20 −100

200

400

600

800

1000

1200

Stress, MPa

Mag

netiz

atio

n, k

A/m

2.02 kA/m2.82 kA/m

Stress

Stress

Field Field

o*

Fig. 4. (a) Comparison of production grade Fe–Ga magnetization versus stress at

2.8 kA/m field obtained from sensing and actuation measurements and (b) minor

loops from alternately varying the field between 2 and 2.8 kA/m and the stress

between 9.4 and 12.6 MPa compression, with the cycle repeated three times.

−20 −10 0 10 20

−1000

−500

0

500

1000


Mag

netiz

atio

n, k

A/m

IncreasingStress

Fig. 5. Magnetization and strain measurements of research grade Fe81.5Ga18.5

hysteresis loop, when the stress is lowered to zero and broughtback to 15 MPa, the magnetization is pushed to point A on the upperbranch of the major loop. Further cycles always return to the upperbranch of the major loop and overlap. These results show that thewidth of the hysteresis loops for applied stress and applied field areconstrained in that knowledge of one determines the other.

3. Model development

A constitutive model relating magnetization and strain to mag-netic field and stress is developed from thermodynamic principleswith stochastic homogenization. A hysteron for single crystallinematerial devoid of imperfections is first derived from thermodynamicprinciples. Stochastic homogenization is then employed to incorpo-rate the smoothing effect of material inhomogeneities.

The first law of thermodynamics states that the rate of changeof the internal energy U is equal to the rate of change of theapplied work plus generated heat, less the heat leaving thesystem. For a thermomagnetomechanical material with stress Tand strain S (using vector notation), magnetic field H, magnetiza-tion M, heat generation Q and heat flux q, this is expressed by

_U ¼ T � _Sþm0H � _MþQ�r � q: ð1Þ

The second law of thermodynamics states that thermal processesresult in entropy Z increase. At temperature y this can beexpressed through the Clausius–Duhem inequality

_ZZ Q

y�r �

q

y

� �: ð2Þ

Eliminating the heat generation by combining the first and secondlaws gives

y _Z� _UþT � _Sþm0H � _M�1

yq � ryZ0: ð3Þ

The dependencies are

U ¼UðS,M,ZÞ, ð4Þ

T¼ TðS,M,ZÞ, ð5Þ

H¼HðS,M,ZÞ: ð6Þ

In practice it is easier to measure magnetization and strain as afunction of field, stress, and temperature. Furthermore, themagnetization can be interpreted as being a function of internalvariables representing r possible domain orientations mk each ofwhich occurs with volume fraction xk. The internal variablesprovide a mechanism for energy dissipation leading to a unifiedhysteresis mechanism for both applied field and stress. The depen-dencies are switched through the Legendre transformation:

GðH,T,yÞ ¼UðM,S,ZÞ�Zy�S � T�m0M �H: ð7Þ

−20 −10 0 10 20−700−600−500−400−300−200−100

0100200


Stra

in×1

06

IncreasingStress

at constant compressive stress (0, 9, 16, 23, 28, 32, 37, 42, and 46 MPa)

−60 −50 −40 −30 −20 −10

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400

600

800

1000

1200

Stress, MPa

Mag

netiz

atio

n, k

A/m

IncreasingField

−70 −60 −50 −40 −30 −20 −10 0−1200

−1000

−800

−600

−400

−200

0

200

Stress, MPa

Stra

in×1

06

Increasingfield

Fig. 6. Magnetization and strain measurements of research grade Fe81.5Ga18.5 at constant magnetic field (1.9, 2.4, 3.2, 4.2, 4.8, 5.6, 6.5, 7.3, 8.1, and 8.9 kA/m).

−40 −30 −20 −10 0

200

300

400

500

600

700

800

900

1000

1100

Stress, MPa

Mag

netiz

atio

n, k

A/m

ConstantField

ConstantStress

Fig. 7. (a) Comparison of research grade Fe81.5Ga18.5 magnetization versus stress

at 4.2 kA/m field obtained from sensing and actuation measurements and

(b) magnetization excursions from the upper and lower hysteresis branches about

a bias of 15 MPa and 4.2 kA/m, cycled three times.


Since magnetization is also dependent on the internal variables mk

and xk, the dependencies after transformation are

M¼MðH,T,y,mk,xkÞ, ð8Þ

S¼ SðH,T,y,mk,xkÞ, ð9Þ

Z¼ ZðH,T,y,mk,xkÞ: ð10Þ

The time rate of change of the free energy G is then

_G ¼@G

@H� _Hþ

@G

@T� _Tþ

@G

@y_yþ

Xr

k ¼ 1

@G

@mk� _mkþ@G

@xk_x

k

" #: ð11Þ

From (7), the time rate of change of the internal energy is

_U ¼ _GþZ _yþy _ZþS � _TþT � _Sþm0H � _Mþm0M � _H: ð12Þ

Substitution of (12) and (11) into (3) gives the following restrictionon thermodynamic processes resulting from changes in the inde-pendent and internal variables:

� Zþ @G

@y

� �_y� m0Mþ

@G

@H

� �_H� Sþ

@G

@T

� �_T

þXr

k ¼ 1

@G

@mk� _mkþ@G

@xk_x

k

" #�

1

yq � ryZ0: ð13Þ

For thermomagnetomechanical materials that are thermodynami-cally reversible, the inequality becomes an equality and the con-stitutive relations can be found from

m0M¼�@G

@H, ð14Þ

S¼�@G

@T, ð15Þ

Z¼� @G

@y, ð16Þ

@G

@xk¼ 0, ð17Þ

@G

@mk¼ 0, ð18Þ

q¼�kry: ð19Þ

Now the following assumptions are made on the processes:

�
Isothermal � Negligible temperature gradients � Reversible domain rotation � Irreversible domain volume fraction evolution
With these assumptions, the following constitutive relationshipsare consistent with the first and second laws of thermodynamics:

m0M¼�@G

, ð20Þ
@H


S¼�@G

@T, ð21Þ

@G

@mk¼ 0, ð22Þ

�@G

@xk

_xkZ0: ð23Þ

For this case, dissipation occurs as domains reconfigure. The freeenergy and volume fraction evolutions need to be defined. Giventhe free energy, the magnetization, strain, and possible domainorientations can be calculated from the constitutive relations(20)–(22). The volume fraction evolution must satisfy theinequality (23) in order for the thermomagnetomechanical pro-cess to satisfy the first and second laws of thermodynamics.

3.1. Energy formulation

The free energy has terms for magnetic anisotropy GA, magne-tomechanical coupling GC, Zeeman or field energy GZ, and elasticstrain energy GE. These energies will be expressed while idealiz-ing the complex domain structure of ferromagnetic materials asa system of non-interacting, single-domain, Stoner–Wohlfarth(S–W) particles. Rather than employ an energy expression whichis valid for all domain variants, a local definition is used for eachvariant which depends only on the easy axis of the variant ck andits temperature dependent anisotropy coefficient Kk

ðyÞ,

GkA ¼

12KkðyÞ9mk�ck92

: ð24Þ

This energy characterizes the torque required to rotate a S–Wparticle away from its easy axis. For small rotations, the coeffi-cient Kk is the slope of the torque-angle curve. For cubic materials,the /100S or /111S directions tend to be easy. Because of crystalsymmetry, the anisotropy coefficient in each direction family is thesame, thus Kk

¼ K100 for all six /100S directions and Kk¼ K111 for

all eight /111S directions. For negative anisotropy coefficients Kk,the direction ck is magnetically hard or an unstable equilibrium. Forthe isothermal processes considered here the anisotropy coeffi-cients are constant.

The total anisotropy energy is the weighted sum of thecontribution from each variant,

GA ¼Xr

k ¼ 1

GkAx

k: ð25Þ

The volume fraction evolution will ensure that using a locally definedanisotropy energy introduces little error into the total anisotropyenergy. For magnetically soft materials, as a domain is pulled awayfrom an easy axis, its size is reduced through domain wall motion andthe size of domains lying nearer an easy axis is increased. Thisdomain reconfiguration is modeled here through the evolution of theS–W particle volume fractions and as xk decreases, the anisotropyenergy contribution from the kth variant is likewise reduced.

The magnetomechanical coupling energy is the weighted sumof the contributions to the mechanical work from the magnetos-triction kk of each particle variant:

GC ¼�Xr

k ¼ 1

ðkk� TÞxk

ð26Þ

and the Zeeman energy is the magnetic work due to each particlevariant

GZ ¼�Xr

k ¼ 1

ðm0MsðyÞm �HÞxk, ð27Þ

where Ms is the magnetization of a S–W particle, constant forisothermal processes. For Galfenol, the magnetostriction of a S–W

particle for cubic materials is used [33]

kk¼

ð3=2Þl100ðyÞðmk1Þ

2


2


2

3l111ðyÞmk1mk

2

3l111ðyÞmk2mk

3

3l111ðyÞmk3mk

1

266666666664

377777777775: ð28Þ

The spontaneous magnetostrictions in the /100S and /111Sdirections are constant for isothermal processes. The mechanicalstrain energy density of the material is

GE ¼�12T � sT, ð29Þ

where the 6�6 compliance s has symmetry consistent with thematerial crystal structure. Summing the energy terms and weight-ing with the volume fractions, the free energy of the material is

G¼Xr

k ¼ 1

xkGkþT � sT, ð30Þ

Gk¼ 1

2Kk9mk�ck92�kk� T�m0Msm

k �H: ð31Þ

From (20), the magnetization is

M¼�1

m0

@G

@H¼Ms

Xr

k ¼ 1

xkmk ð32Þ

and from (21) the strain is

S¼�@G

@T¼Xr

k ¼ 1

xkkkþsT, ð33Þ

thus to calculate the magnetization and magnetostriction, the S–Wparticle orientations and volume fractions need to be known.

3.2. Calculation of particle orientations

The magnetic orientations mk of the S–W particles are calcu-lated from the minimization (22). This minimization is con-strained since the vector mk is a unit vector. The constraint isC ¼ 9mk9�1¼ 0. The constrained minimization can be cast in theform of an inhomogeneous eigenvalue problem by using Lagrangemultipliers, @Gk=@mk ¼ g@C=@mk. Gathering terms from (31) andexpressing the particle free energy as Gk

¼ 12 mk � Kkmk�mk � Bk

the eigenvalue problem is

ðKk�gIÞmk ¼ Bk, ð34Þ

where the magnetic stiffness matrix Kk and force vector Bk are

Kk¼

Kk�3l100T1 �3l111T4 �3l111T6

�3l111T4 Kk�3l100T2 �3l111T5

�3l111T6 l111T5 Kk�3l100T3

264

375, ð35Þ

Bk¼ ½ck

1Kkþm0MsH1 ck

2Kkþm0MsH2 ck

3Kkþm0MsH2�

T: ð36Þ

While the orientation can be easily solved for in terms of gthrough diagonalization, determination of g requires solution of asixth-order polynomial obtained by substituting the g dependentorientation into the constraint. It is important to keep thecomputation expense of the hysteron to a minimum since a largenumber of hysterons are summed in the stochastic homogeniza-tion process. To this end, the constraint is relaxed throughlinearization about the easy direction ck. This is accurate becauseas a field or stress pulls a particle away from the easy direction, ata critical energy level the particle flips to an equilibrium closer tothe applied field or perpendicular to the applied principal

−5 0 5

−1

−0.5

0

0.5

1


Rel

ativ

e M

agne

tizat

ion

00

300

600

900

Fig. 8. Anisotropic hysterons.


stresses. This is modeled through an instantaneous change in xk.Hence particles oriented near an easy axis always have largevolume fractions as compared to particles which have beenrotated far from their easy axis. For the relaxed, linear constraint,the solution to the inhomogeneous eigenvalue problem is

mk ¼ ðKkÞ�1 Bk

þ1�ck � ðKk

Þ�1Bk

ck � ðKkÞ�1ck

ck

" #: ð37Þ

The particle orientations (37) define the possible states of a 3D,anisotropic, field and stress dependent magnetization and strainhysteron. The magnetization of the hysteron is calculated from (32)and the strain from (33). An evolution equation for xk will bedefined in the next section. This evolution defines the switchingof the hysteron states and is consistent with the second law ofthermodynamics.

3.3. Evolution of domain volume fractions and hysteron

development

The hysteron represents the magnetization and strain at thedomain level and is constructed by tracking a single S–W particleas it switches state from an initial orientation variant mI to a finalorientation variant mF . This switching is modeled throughchanges in xk where prior to switching xI

¼ 1 and xk¼ 0 for ka I

and after switching xF¼ 1 and xk

¼ 0 for kaF. Switching occurswhen the energy difference GI

�GF : ¼ GIF reaches a coerciveenergy Ec. As the field and stress change, the energy differencewith the globally minimum orientation always reaches thecoercive energy first, hence the state of the hysteron after switch-ing mF is always the globally minimum orientation variant. Sincexk¼ 1 for k¼ I or k¼F and all other xk are zero, from (30) the free

energy of the hysteron is

G¼ xIGIþxFGF

�s � T ð38Þ

and the thermodynamic condition (23) which is the rate of energydissipation becomes

�@G

@xI_x

I�@G

@xF_x

F¼�GI _x

I�GF _x

FZ0: ð39Þ

The criterion for switching from mI to mF can be expressedmathematically as a dirac-delta function:

_xF¼ dðGIF

�EcÞ, _xI¼� _x

F: ð40Þ

The energy dissipation becomes

�GI _xI�GF _x

F¼ GI _x

F�GF _x

F¼ GIF _x

F¼ GIFdðGIF

�EcÞ ¼Ec , GIF

¼ Ec

0, else:

(

ð41Þ

Thus there is an instantaneous energy dissipation of Ec whenswitching occurs. A hysteron for changing field or stress isconstructed as follows. For each time step, the orientations mk

are calculated from (37) and the particle energies Gk from (31).The initial state k¼ I must be known. The final state k¼F is takenas the globally minimum state. After a step change in the field orstress, the hysteron state either stays as k¼ I or switches to k¼F ifthe switching criterion is met. The magnetization and strain of thehysteron are then calculated from (32) and (33), in other words,the magnetization before and after switching is MsmI and MsmF

and the magnetostriction before and after switching kI and kF

where kk is given by (28) for cubic materials.

3.4. Reduction of vector hysteron to scalar hysterons

The classical Preisach model is scalar and depends on magneticfield only. The Preisach relay switches between two field independent

states, 71 representing domains as oriented up and down.Switching occurs at a field value of H¼ 7Hc where Hc is thecoercive field and the magnetization contribution of the relay is7Ms. For the hysteron presented here, consider a material withuniaxial anisotropy having easy axes in the ½7100� directions.Associated with these two easy axes are two variants of S–Wparticle orientations, m½100� and m

½100� having the same anisotropyconstant, K100. If the field is applied in the direction u¼ ½1 0 0�, thenthe hysteron reduces to the classical Preisach relay. From Eq. (37),the component of the variant m½100� is þ1 in the direction of u,i.e. u �m½100� ¼ 1 and the component of the variant m

½100� is �1.No rotation of the S–W particles occurs with field applicationbecause it is applied along the easy axes. The energies of the twovariants are 8m0MsH�1=2K giving the energy difference betweenthe two variants as G

½100��G½100� ¼ 2m0MsH. Therefore the hysteronswitches from the ½100� direction to the ½100� direction at thecoercive field Hc ¼ Ec=ð2m0MsÞ.

Smith’s homogenized energy model [28] employs two energy-based hysterons—one for static operation and another which hasrate-dependence characterizing the thermal after-effect . Thestatic hysteron differs from the Preisach relay in that up anddown states vary linearly with field. This same relay behavior canbe achieved with the proposed hysteron here by again consider-ing uniaxial material but applying the field away from the easyaxes. Examples are shown in Fig. 8 with the field applied 01, 301,601, and 901 from the easy axis in the ½100� direction. At anincreasing angle the hysteron reflects greater difficulty in mag-netizing since the direction is further away from the easy axes. At901 the two variants have the same orientation and energy and noswitching occurs. For an angle y the slope or relative suscept-ibility of the up and down states is

w¼ sin2ðyÞ

m0Ms

K100ð42Þ

and the remanence is 7 cos y. The energy difference is

G½100�� G½100� ¼ 2 cosðyÞm0MsH which results in a coercive field for

switching from the down to the up state Hc ¼ Ec=ð2 cosðyÞm0MsÞ:

This anisotropic hysteron illustrates an important differencebetween Preisach models and homogenized energy models. Bothemploy an elementary hysteron with statistically distributed para-meters, however, the former includes effects from additionalphysics such as anisotropy or magnetomechanical coupling inthe density functions while the latter includes additional physicsin the hysteron, which describes the underlying material behavior.Typically this results in simpler density functions, or in thediscretized model, fewer weights.


3.5. Description of measurements with elementary hysteron

The hysteron can describe the underlying shape of the mag-netization versus field curves in Figs. 2(a) and 5(a) as well as themagnetization versus stress curves in Figs. 6(a) and 3(a) and (b).Even though the measurements are unidirectional, a vector modelis needed for their description because anisotropic, 3-D domainrotation occurs.

For /100S oriented, research grade material a /100S orienta-tion is aligned with the rod axis which is the direction of field andstress. The stress dependence of the hysteretic burst regions inFig. 5(a) can be understood from the hysterons in Fig. 9(a). Thelow-field linear region is due to rotation of /100S domainsperpendicular to the field direction. The hysteron represents thisregion through the field and stress dependence given by (37)of mk which have easy axes perpendicular to the field. There are foursuch mk which explains why the hysteron has only three uniquestates when calculating magnetization and strain in a /100Sdirection. The burst region occurs by domain reconfiguration asthe size of domains along the /100S direction aligned with thefield grows at the expense of the perpendicular domains. In thehysteron, the burst region is described by switching from mI witheasy axes perpendicular to the field to mF with the easy axisparallel to the field. The coercive or dissipation energy Ec causes adelay which accounts for the noticeable hysteresis in the burstregion and lack thereof in the low-field region where themagnetization process is dominated by domain rotation. Withincreased stress, the energy of mk with easy axes perpendicular tothe field is decreased, thus the field location where switchingoccurs is pushed to higher fields.

The additional hysteretic region at low fields for the produc-tion grade, field applied major loops results from misalignment.Two of the /100S easy directions near the plane perpendicular to

−15 −10 −5 0 5 10 15

−1000

−500

0

500

1000


Mag

netiz

atio

n, k

A/m

IncreasingStress

Fig. 9. Magnetization of a hysteron in the ½100� direction for (a) appli

−20 −10 0 10 20−1500

−1000

−500

0

500

1000

1500


Mag

netiz

atio

n, k

A/m

Fig. 10. Hysteron calculations of (a) magnetization and (b) strain

the rod axis are closer to the negative rod direction and the othertwo are closer to the positive rod direction. Thus hystereticswitching occurs between these orientation variants as the fieldcrosses zero (see Fig. 10(a)). Although this switching results inenergy loss, it does not result in a low-field hysteresis region forthe magnetostriction (see Figs. 2 and 10) because magnetostrictionis a quadratic relation; the variants which are slightly closer to thepositive field direction and the variants which are slightly closer tothe negative field direction have the same magnetostriction.

The center of the burst region in the magnetization versus fieldmeasurements can be calculated analytically from GI

¼ GF and thesusceptibility in the low field, domain rotation region, can becalculated from (37). The anisotropy coefficient K100 can be deter-mined directly from the measurements since (37) is an analyticalfunction of K100 . The particle magnetization Ms and the magnetos-triction l100 can be directly found from the saturation magnetiza-tion and magnetostriction values. The shear magnetostictioncoefficient l111 does not affect ½100� behavior because no shearstresses are present. The coercive energy can be found from theareas of the hysteretic regions which give the energy loss. Thevalues for the hysteron parameters are in Table 1.

Magnetization versus stress loops in Figs. 6(a) and 3(a) and (b)can be understood from the hysterons in Fig. 9(b). Stress causesdomain switching from the bias field direction or high state to theperpendicular plane or low state. At low stresses, when thematerial has been saturated positively prior to application ofthe bias field, if the hysteron is double-valued, it will be at thehigh state. Application of stress switches the hysteron to the lowstate and upon removal of the stress it remains in the low state. Insubsequent loops, the hysteron starts and ends in the low state.This accounts for the observed non-closure of the first loop andclosure of all subsequent loops; it is due to the history of thedomain configuration. When the material is saturated negatively

−70 −60 −50 −40 −30 −20 −10 00

200

400

600

800

1000

1200

1400

Stress, MPa

Mag

netiz

atio

n, k

A/m

IncreasingField

ed field at constant stress and (b) applied stress at constant field.

−20 −10 0 10

−1000

−800

−600

−400

−200

0


Stra

in×1

06

IncreasingStress

in the near ½100� direction for applied field at constant stress.

Table 1Hysteron parameters for production grade Fe81.4Ga18.6 and research grade Fe81.5Ga18.5.

Grade m0Ms (T) K100 (kJ/m3) Ec (J/m3)

Research 1.55 35 218

Production 1.6 35 192

l100�106 l111�106 E (GPa)

Research 177 N/A 76.5

Production 110 3 98

−15 −10 −5 0 5 10 150

5

10

15

20


Gib

bs F

ree

Ener

gy, k

J

Increasingstress

−15 −10 −5 0 5 10 150

5

10

15

20

Magnetic Field, kA/mG

ibbs

Fre

e En

ergy

, kJ

Increasingstress

Fig. 11. Dependence of Gibbs free energy on ½100� field and stress for the Fe81.6Ga18.4, production grade sample, calculated from (a) the measurements and (b) the model

with a single hysteron; compressive bias stresses are 0, 19, 35, 52, and 70 MPa.

−15 −10 −5 0 5 10 150

5

10

15

20


Gib

bs F

ree

Ener

gy, k

J

Increasingstress

−15 −10 −5 0 5 10 150

5

10

15

20


Gib

bs F

ree

Ener

gy, k

J

Increasingstress

Fig. 12. Dependence of Gibbs free energy on ½100� field and stress for the Fe81.5Ga18.5, research grade sample, calculated from (a) the measurements and (b) the model with

a single hysteron; compressive bias stresses are 0, 16, 28, 37, and 46 MPa.


prior to bias field application, the hysteron starts in the low stateand hence the first loop is closed also. A departure from thispattern occurs when the bias field is zero. In this case thehysteron begins negative, described by the mI with easy axisoriented in the negative field direction. Applied stress switchesthe hysteron state to mF with easy axis perpendicular to the biasfield. Further cycles do not result in magnetization changebecause there is no field causing a preference for either thepositive or negative rod directions. Thus the remanent magneti-zation of materials with kinematically reversible magnetomecha-nical coupling can be completely eliminated with a single stresscycle which forces all domains to the perpendicular plane. Forsufficiently high bias fields, the hysteron is not double-valued atzero stress and thus there is no distinction between saturating thematerial positively or negatively; in both cases, the starting andending domain configuration is all domains aligned in the biasfield direction.

The dependence of the Gibbs free energy on ½100� directed fieldcan be calculated from the measurements by numerically inte-grating (20); the free energy calculated from applied field mea-surements is compared with the free energy of a hysteron inFigs. 11 and 12 for the research and production grade samples,

respectively. Just as a single Preisach relay is blocky compared tomagnetization-field loops, the energy calculated from a singlehysteron has much sharper transitions than the measurements.The energy loss associated with hysteretic domain reconfigura-tion causes a difference between the beginning and ending energylevel which is characterized in the hysteron through the para-meter Ec. For a full major loop of the research grade material, thehysteron switches four times and for the production gradematerial six switches occur, i.e. two switches occur for eachhysteresis region—one on the up-side and one on the down-side . The five energy versus field curves for the research gradesample (calculated at different bias stresses) have nearly the sameenergy loss with an average of 873 J/m3 which gives a value of218 J/m3 for Ec (a fourth of the loop energy loss.) The five energyversus field curves for the production grade sample have anaverage energy loss of 1,149 kJ/m3 giving a value of 192 J/m3 forEc (a sixth of the loop energy loss.) Though the estimation of Ec isless for the production grade sample, the total energy loss isgreater since more switching occurs owing to the low fieldhysteresis region. Despite this, the comparison is still unexpectedand may result from describing the data with a single hysteron.In the next section, stochastic homogenization is employed where

−5 0 5−1

−0.5

0

0.5

1


Rel

ativ

e M

agne

tizat

ion

0o

30o

60o

90o

Fig. 14. Stochastic homogenization of the anisotropic, vector magnetization model.


statistical variation in Ec and the magnetic field are considered.Then, a collection of hysterons is considered and the energy lossin a major loop is the average value of the product of the numberof switches occurring in a hysteron and Ec.

The dependence of the Gibbs free energy on ½100� directedstress can be calculated from the measurements by numericallyintegrating (21). A comparison between the measurements andthe free energy of a single hysteron is shown in Fig. 13 at fourdifferent bias fields. Stress–strain data for the production gradesample is not available. For the research grade sample, threestress cycles were performed starting at zero stress at each biasfield. Energy is lost for each cycle resulting in a downward shift ofthe energy of an amount 2Ec since two switches occur—the upand the down switches between the directions parallel to the fieldand near perpendicular to the field. The average loss per cycle forthe four measurements is 659 J/m3 which is less than for appliedfield major loops. That the loss per loop is less than for the appliedfield measurements is expected since fewer switches or domainreconfigurations occur. According to the model the loss per loopshould be exactly half since half as many switches take place.Since it is not exactly half, this suggests the presence of a smallamount of energy loss from purely mechanical hysteresis which isneglected here where the purely mechanical strain is modeledthrough Hooke’s law or a quadratic strain energy density.

3.6. Stochastic homogenization

Material inclusions and lattice imperfections cause variationsin the local magnetic field and coercive energy [28]. This variationis modeled using a statistically distributed interaction field HI ,superimposed on the macroscopic applied field H. The macro-scopic magnetization and strain can be calculated through sto-chastic homogenization of the interaction field and coerciveenergy:

MðH,TÞ ¼

Z 10

Z 1�1

MðHþHI ,T,EcÞnðHI ,EcÞ dHI dEc , ð43Þ

SðH,TÞ ¼

Z 10

Z 1�1

SðHþHI ,T,EcÞnðHI ,EcÞ dHI dEc : ð44Þ

The homogenization procedure (43) is similar to the homoge-nized energy model of Smith et al. [28] when thermal activation isneglected. Different energy potentials are used for the hysteronsand the energy formulation here is done in 3-D . Additionally, acoercive energy consistent with the second law of thermody-namics was defined rather than a coercive field. This homogeni-zation procedure has a distinct computational advantage overthe vector implementation of Preisach models. Vector Preisachmodels for 3-D magnetization-field behavior use six statistically

−60 −50 −40 −30 −20 −10 00

10

20

30

40

50

Stress, MPa

Gib

bs F

ree

Ener

gy, k

J

Increasingfield

Fig. 13. Dependence of Gibbs free energy on ½100� field and stress for the Fe81.5Ga18.5, re

a single hysteron; bias fields are 1.9, 3.2, 6.5 and 8.9 kA/m.

distributed parameters, two for each dimension which dictatewhen switching occurs from up to down and visa versa. Themodel proposed here includes four statistically distributed para-meters meaning that the model implementation requires quad-ruple integration rather than the sextuple integration required forvector models using the classical Preisach model.

The procedure here offers the additional advantage of simplerdensity functions since the effects of anisotropy and stress areincorporated in the hysteron behavior rather than the densities.Consider again the case of uniaxial anisotropy for which thehysterons are shown in Fig. 8. Since the physical informationregarding anisotropy is embedded in the hysteron, a simpleGaussian distribution of each component of the interaction fieldcan be used. Fig. 14 shows the homogenized model with aGaussian distribution; the behavior is anisotropic even thoughall three components of the interaction field have the samestandard deviation. An exponential distribution was used for thecoercive energy.

3.7. Comparison with experiments

Here, the homogenized energy model is compared with researchgrade measurements. The hysteron parameters in Table 1 are used.The discretized probability density in the homogenized energy model(43) is found through least-squares optimization to a single M2Hcurve, measured with a 23 MPa compressive bias stress. Theprocedure for identifying a general density, described in [34], is used.The integrals are performed over small segments using Gauss-quadrature . The bounds of the integrals for the interaction field arethe fields required for positive and negative saturation, 710 kA=m.The bound for the coercive energy was chosen as 600 J/m3, over twice

−60 −50 −40 −30 −20 −10 00

10

20

30

40

50

Stress, MPa

Gib

bs F

ree

Ener

gy, k

J

Increasingfield

search grade sample, calculated from (a) the measurements and (b) the model with

0 2 4 6 80

200

400

600

800

1000

1200


Mag

netiz

atio

n, k

A/m

−20 −18 −16 −14 −12

Stress, MPa

dashed, modelsolid, data

Fig. 16. Magnetization excursions from the upper and lower hysteresis branches

about a compressive bias of 15 MPa and 4.2 kA/m, cycled three times.


the value used for a single hysteron in Section 3.5. This bound waschosen so that the value of nðHI ,EcÞ is approximately zero at thebounds. This condition is necessary to ensure that the error intro-duced by using finite integral bounds is minimal. The Gauss quad-rature rules give the hysteron evaluation points ðHIÞ1,i, ðHIÞ2,j, ðHIÞ3,k,Ec,l and weights wijkl. The hysteron values are then calculated at eachof the Gauss points MðHþHI ,T,EcÞijkl. The values of the density ateach Gauss point, nijkl are found through the least-squares optimiza-tion routine lsqnonlin in Matlab. The objective function for theroutine is the error between the model and the measurements. Themodel value of magnetization for each applied field and stress valueare calculated with the summation over the four indices,MðHþHI ,T,EcÞijklðnwÞijkl, which approximates the fourth-order inte-gration. The strain is calculated similarly, but not used in theoptimization scheme.

To illustrate the accuracy and properties of the model, thedensity is first found from optimizing for the error between themodel and the measured magnetization values for applied field ata constant compression of 23 MPa. A comparison with the modelis then made between the measured strain of this same experi-ment along with the magnetization and strain of a separateexperiment of applied stress at a constant magnetic field of4.8 kA/m. The result is displayed in Fig. 15. The density is onlyoptimized for the measurements in Fig. 15(a), yet there isexcellent agreement between the model and the measurementsin Fig. 15(b) and (c). This confirms that the hysteron describes theunderlying physical behavior since from figure to figure, the

−10 −5 0 5 10

−1000

−500

0

500

1000


Mag

netiz

atio

n, k

A/m

Model

Data

−60 −50 −40 −30 −20 −100

200

400

600

800

1000

1200

Mag

netiz

atio

n, k

A/m

Stress, MPa

Model

Data

Fig. 15. Model and measured magnetization and strain from (a), (b) applied field at

density remains the same and only the hysteron changes. Thoughthe density was found from measurements for applied field atconstant stress, the model accurately describes the measurementsfor applied stress and constant field. Fig. 16 shows that thehysteresis properties of the model agree with the measurements.Starting from the lower branch of a magnetization-field majorloop, a cyclic reduction in the stress pushes the magnetization tothe upper branch in a single cycle with subsequent cycles

−10 −5 0 5 100

50

100

150

200

250


Mic

rost

rain

Model

Data

−60 −50 −40 −30 −20 −10

−800

−600

−400

−200

0

200

Mic

rost

rain

Stress, MPa

Model

Data

a compression of 23 MPa, and (c), (d) applied stress at a bias field of 4.8 kA/m.


returning to the upper branch. Starting from the upper branch, acyclic increase in the stress pushes the magnetization to the lowerbranch with subsequent cycles returning to the lower branch. Thecoupled nature of the hysteresis as well as the nonclosure ofinitial loops is described by the model through the history ofmagnetic domain orientations.

4. Concluding remarks

Measurements were presented to characterize the couplednonlinear and hysteretic magnetization and strain of productionand research grade Galfenol due to applied stress and field. It wasshown that hysteresis for both applied quantities can be attrib-uted to the same physical mechanism and that major magnetiza-tion versus stress loops in compression depend heavily onmagnetic history at low stress levels. Remarkable reversibility inthe magnetomechanical coupling was demonstrated by generat-ing the same magnetization versus stress hysteresis loop bothfrom a series of constant stress experiments and from a singleconstant field experiment. Cyclic application of alternatelyapplied field and stress did not result in any noticeable accom-modation, i.e. Galfenol constitutive behavior is kinematicallyreversible.

A thermodynamic framework, satisfying the first and secondlaws of thermodynamics, was developed to describe the observednonlinear and hysteretic behavior. Like the measurements, mag-netization and strain calculations from the model are thermo-dynamically irreversible and kinematically reversible. Hysteresiswas attributed to energy loss during the process whereby thevolume fractions of differently oriented domains change. Thisprocess was modeled by tracking the orientation of a number ofelementary hysterons whose states represent the energeticallypossible domain orientations. The hysteron is 3-D, anisotropic,and both stress and field dependent. The model thereby does notdepend on complex density functions to describe these effects, asdo models based on the classical Preisach model. An additionaladvantage over Preisach models is that hysteron switching ischaracterized by a coercive energy rather than a 3-D coercivefield. As a result, the integration order is four rather than six. Themodel provides a physical and accurate description of 3-Dmagnetic and strain hysteresis for anisotropic magnetostrictivematerials.

Acknowledgments

We wish to acknowledge the financial support by the Office ofNaval Research, MURI grant #N000140610530. Measurements wereperformed at the Naval Surface Warfare Centery, Carderock Divisionas part of the Naval Research Enterprise Internship Program forP.G.E., with Marilyn Wun-Fogle and Dr. James Restorff as mentors.

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