IEEE TRANSACTIONS ON MAGNETICS, VOL. 45, NO. 1, JANUARY 2009 83

The Pursuit of Hysteresis in Polycrystalline FerromagneticMaterials Under Stress

Daniel Peter Bulte

Physics Group, FMRIB Centre, University of Oxford, Oxford OX3 9DU, U.K.

External stresses alter the magnetic properties of ferromagnetic materials such as iron and steel, a fact that has been the basis ofsubstantial study in nondestructive testing. Existing theories and models have so far not proven reliable or accurate enough to developa practical means of using the developed theory relating stress and magnetization to measure biaxial strains without prior knowledgeof the strain or magnetic history of the sample. A deterministic model of ferromagnetic hysteresis and the effects of external stressesin materials such as iron and steel is introduced by this study. Changes in hysteresis loops due to stress are explained via changes inthe magnetocrystalline anisotropy at the crystal-unit level, and are extended to the macroscopic effects that are seen in experiments.An original equation is presented which accurately describes experimentally acquired major hysteresis loops and directly relates twoparameters to the two perpendicular principal strain axes thereby providing a technique able to determine the absolute stress/strainexperienced by the sample. This model will potentially enable quantitative, nondestructive stress measuring devices to be developed.

Index Terms—Hysteresis loops, pursuit curve, strain, stress.

I. INTRODUCTION

T HE hysteretic nature of magnetization curves of ferro-magnetic materials is one of their key features (Fig. 1).

The source of this hysteresis has been the subject of enduringdebate over the last century and many significant advanceshave been made in understanding the interactions betweenthe quantum, electromagnetic, and chemical aspects of thebehavior. An explanation of the effect which allows an obviouspredictive model to be produced has, however, been quite elu-sive. Models such as the those presented by Jiles and Atherton[1], Stoner and Wolhfarth [2], and Preisach [3] have had suc-cesses in explaining many aspects of hysteresis and have greatlyimproved our understanding of the magnetization process inferromagnetic materials. However, all of the models previouslyproposed are in some way limited in their applicability andoften require additional corrections in order to reconcile themwith experimental data (see Liorzou et al. [4] for a detailedreview).

The interaction between externally applied stresses and themagnetic susceptibility of a ferromagnetic material is a subtleyet fundamental aspect of ferromagnetics. Whereas the magne-tomechanical effect considers the condition under which the ap-plied magnetic field is held constant while the applied stressesare varied, in this work we consider primarily the inverse of thiswhere the applied stress is constant and the applied magneticfield is varied. The nature of the connection between appliedexternal stresses and the anisotropic susceptibility of polycrys-talline ferromagnetic materials has previously been examined[5]. The existence of so called coincidence points on major andsymmetric minor hysteresis loops of stressed samples of ferrousmaterials has also been shown experimentally [6]–[9] and theirrelevance to interpreting changes in ferromagnetic hysteresisdiscussed [5]. An overlay of multiple major hysteresis loops

Manuscript received April 20, 2007; revised October 01, 2008. Current ver-sion published January 30, 2009. Corresponding author: D. P. Bulte (e-mail:bulte@fmrib.ox.ac.uk).

Digital Object Identifier 10.1109/TMAG.2008.2007510

Fig. 1. Major hysteresis loop for a sample of unstressed, annealed mild steel atroom temperature.

in steel, with approximately the same maximum applied field(Fig. 2), clearly shows that in the second and fourth quadrantsthere are points through which all of the loops pass, regardlessof the applied uni- or bi-axial stresses. This effect was theorizedto be due to all magnetic moments within the material being ineither 180 domain walls or easy crystalline directions at thesefield strengths, thereby rendering the magnetization essentiallyunchangeable by structural forces [5], [9].

Other theories which have been formulated to describe stress/magnetization behavior using either hysteresis or the magne-tomechanical effect [1], [4], [10]–[18] rely on such mechanismsas structural imperfections in the crystal lattice, which cause do-main wall pinning, or Barkhausen noise and thereby energy lossaround the hysteresis loop. General ferromagnetic hysteresistheories [3], [19]–[22], although simple to work with, either donot reflect real ferrous materials, or do not resemble experimen-tally obtained hysteresis curves. The most accurate theory thusfar has been achieved by obtaining information about the ma-terials investigated from their anhysteretic curves [23]. Thesemodels have been shown to be very robust and reliable. How-ever, they still tend to deviate from experimental data in situ[24], and require the prior acquisition of numerous parametersand the anhysteretic data itself, which is a nontrivial procedure[25].

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84 IEEE TRANSACTIONS ON MAGNETICS, VOL. 45, NO. 1, JANUARY 2009

Fig. 2. Major hysteresis loops for a single piece of steel under different biaxialstress conditions ranging from �120 MPa to 120 MPa clearly showing the 2coincident points.

Schneider defines a process in which field is held constantand stress varied an process; accordingly when stress isheld constant, and the field varied, then the process is calleda process [26]. Investigations into stress/magnetizationand energy/hysteresis relationships have primarily considered

arrangements; however, notable work has investigated andattempted to model other variants of these interactions. Adlyand Mayergoyz have theorized an interaction derived from aPreisach model [27]. Sablik et al. have investigated the effectof a noncoaxial application of a uniaxial stress and field [28], aspecial subset of the biaxial stress conditions considered herein.The energetics of hysteresis processes have been investigatedby Hauser [29], [30] and more recently the influence of appliedstresses on these was considered by Smith et al. [31], [32] inrelation to seminal work by Pitman [18].

The model which was first presented by Bulte and Langman[5] required expansion to include a description of the behaviorat the crystal unit level, as well as a practical, robust equationwhich would relate the coincident points of a given material toa hysteresis loop of that material while experiencing a given ex-ternally applied stress. For the model to be applicable to all cir-cumstances in polycrystalline ferromagnetic materials it needsto explain the behavior of flux density versus magnetic field dataat all levels, from a single, ideal crystal unit, up to a macro-scopic, multigrain, multidomain sample of the material.

II. MACROSCOPIC MODEL

A bulk sample of iron or steel will contain many millions ofgrains; each grain consists of a small volume in which the ionsare arranged in a body-centered cubic crystal lattice (the modelis also applicable to other lattice structures with appropriatechanges for the different crystalline arrangements). The mag-netocrystalline anisotropy within each of these grains is wellunderstood; however, the way in which these many millions ofgrains interact to produce the effects measured from the bulkmaterial has yet to be explained satisfactorily.

In a bulk sample of iron or steel we can assume that there is aspherically symmetric, isotropic distribution of lattice directionsdue to the multitude of grains present. At saturation all of the

magnetic moments within the material will be aligned and thevector sum will be equal to the sum of the magnitudes of all ofthe moments, . As the field strength is reduced the momentswill begin to rotate away from the direction of the applied field.All changes in magnetization occur as a result of the rotation ofmagnetic moments, even the domain-wall motion is a movingwave of rotations, thus the changes can be considered as eithersome or all of the magnetic moments rotating towards or awayfrom the applied field.

Reduction of the applied field will result in a spatial de-phasing of the moments as they rotate to directions within theirregions which are determined by the combination of the localfield (and structural) environment, and the externally appliedfield. The bulk of the change in flux density which occursduring the progression around a major hysteresis loop is almostentirely due to the movement of 180 domain walls, and therotation of moments from one external field direction to theother. The formation of closure domains and movements of90 walls enable this process to occur with minimal energy;however they do not themselves result in substantial changesin the net flux density of the material. Near saturation, changesare dominated by rotation; at low fields domain wall motionhas more effect.

Assuming a homogeneous and continuous distribution oflattice directions is to be found in a macroscopic, multigrainsample of ferrous material, the evolution of the magneticmoments from saturation in one direction to saturation in theopposite direction can be considered. As the magnitude of thefield decreases the moments will begin to rotate away from thefield direction in order to maintain equilibrium between the netfield experienced and the anisotropy energy of their directionof orientation. As moments reach lattice (easy) directions theywill effectively bind to these directions until enough energyis present to overcome this binding; in some cases they willform domains as groups of moments bind to the same direc-tion within regions of regular lattice. The exact nature of theinteraction between hysteresis and the energy associated withbinding to lattice directions is the subject of a forthcomingpublication by the same author. These bindings will beginalmost immediately upon reducing the field from the saturationvalue, as many easy axes will be very close to the applied fielddirection, making the changes nonreversible. In some cases,a 180 domain wall will form as the field decreases and thiswill gradually move across the domain. In other cases, the“nearest easy axis” will be far enough away from the appliedfield (up to 54.74 ) that even the remnant field will be sufficientto maintain a degree of magnetization for some moments,effectively holding them rotated away from easy axes. Betweenthe saturation field and zero field the sample will generallyconsist of a combination of moments in domains, moments indomain walls and moments that are rotated away from easyaxes. If the applied field direction is reversed and increasedfrom zero, the unbound moments will rotate further towardseasy directions until the critical field strength is reached(which corresponds to the “binding energy” of the lattice). Atthis point the major hysteresis loop of the material becomesindependent of applied stress. Further increases in the fieldstrength in this direction will now cause the moments to unbind

BULTE: PURSUIT OF HYSTERESIS IN POLYCRYSTALLINE FERROMAGNETIC MATERIALS UNDER STRESS 85

from the easy directions and start to spatially rephase in thenew field direction. This “binding” to easy directions is mosteasily seen in the very “square” hysteresis loops seen in idealmodels of a single domain with the applied field parallel toan easy direction such as the Preisach and Stoner-Wolhfarthmodels. Under such circumstances it is apparent that reversalof the field below some critical magnitude produces no changein magnetization, and thus no rotation of the moments. Oncethe critical field is reached the binding is overcome and all ofthe moments suddenly rotate towards the field direction.

Minor hysteresis loops, asymmetrical loops and the normaland anhysteretic curves are all results of domain wall motion andlattice binding events which prevent the moments from freelymoving when the field is altered.

III. STRESS-INDUCED ANISOTROPY

In practice, it is much easier to measure the magnetic fluxdensity than the magnetization, although the conversion is notparticularly difficult. To simplify the practical adaptation of ob-taining a measurement of strain or stress from a ferromagneticmaterial using this theory an equation was sought which directlylinked single values for each of the principle stress/strain axesto the shape of a hysteresis loop obtained from the bulk sample.By limiting ourselves to elastic strains we can define a satura-tion point as the field which would saturate the sample at theplastic limit, and the sample’s flux density at that point. The lo-cations of the coincident points on such loops are constants ofthe material, and thus all that is required is a description of thecurve between coincidence and saturation.

The application of an external stress to a sample of ferro-magnetic material alters the shapes of hysteresis loops obtainedexperimentally. From a structural perspective, the actual lat-tice spacings and angles change very slightly within a perfectcrystal when it is strained. In a macroscopic sample the indi-vidual grains may slightly alter their relative orientations, sepa-rations and shapes as a result of the bulk distortions caused bythe stress. These changes constitute the strain on the material.The interaction between stress and magnetization can be visu-alized as a physical change in the separation of the atomic sitescausing changes in the magnetocrystalline anisotropy energiesassociated with each direction within the crystal. In fact, the rel-ative angles between the atomic sites will change very slightlyas a result of this strain; however, the anisotropy energies as-sociated with intermediate directions can change substantially.Thus, at the single crystal unit level, there is an analogy betweenstrain and the inherent magnetocrystalline anisotropy. A smalldistortion of the lattice can be compared to a small rotation ofthe field relative to the lattice directions.

The shape of a hysteresis loop is very strongly dependent onthe anisotropy energy associated with a given direction of ap-plied field. Thus changes in the anisotropy of a sample of crystallattice will affect the shape of the hysteresis loop accordingly.Within each grain the anisotropy energies associated with givendirections will change as the lattice is strained; this will result inthe rotation of individual magnetic moments and thus a changein the net magnetization as more moments are either rotated to-wards or away from the direction of the field and/or measure-ment. However, moments that are bound to lattice (easy) direc-

tions will not rotate away from the lattice; this is what producesthe coincident points as all moments are effectively bound atthese points.

It is not feasible to average the multitude of magnetic mo-ments or even the net magnetic vectors of the grains or domains;however due to the restrictions placed upon their movementsand knowledge of the coincident points, a mathematical methodof obtaining the same result is achieved via a technique origi-nally derived for theoretically plotting the optimal predator/preypursuit trajectories [33]. If we constrain the permitted relativemotions of the predator and prey, and include an accelerationterm for the predator, we can construct an equation which isequivalent to the averaged net response of the moments to thechanging applied field.

A general curve of pursuit is given by

(1)

where if moves along a known curve, then describes a pur-suit curve if is always directed toward , and and movewith uniform velocities [34]. If the velocities are not uniform,

is restricted to motion in a straight line, and the region isbounded by , and , then the relation-ship between the predator and the prey can be described by

(2)

where and are constants relating to the relative velocities ofthe predator and prey.

In ferromagnetics the prey is the applied field and the predatoris the flux density which will alter its speed (i.e. change slope) asthe field increases (runs away). This may seem to be the oppositeof what would be expected; however, it is the fox that changesits direction to chase the rabbit if the rabbit runs in a straightline perpendicular to their original direction of separation, justas the flux density changes in response to the field running in astraight line. On a plane defined by , and

, the field “runs” from to and theflux density will pursue it from its starting point atto the point where it “catches its prey” at . This pursuittrajectory is given by

(3)

which can be rearranged to give

(4)

where and , with and the satu-ration point values, and the coincident point values, and

and are the biaxial strain parameters. The strain pa-rameters are given by exponentials which relate them directly to

86 IEEE TRANSACTIONS ON MAGNETICS, VOL. 45, NO. 1, JANUARY 2009

Fig. 3. Comparison of the model presented in Equation (3) with an experi-mental data set obtained from mild, structural steel with no applied stresses.The curve shown is the upper quarter of a hysteresis loop between the coinci-dent point at�� and the positive saturation point. The values of � and � weredetermined using a least-squares method, and the model fit has an � � �����.

the applied stresses. These parameters are material dependent,and were observed empirically to have the form

(5)

where and are the longitudinal and transverse stresses uponthe material, the exact forms being material dependent. If thevalues for and are in the second quadrant, the upperquarter of the loop will be described between the values ofand . If the values are in the fourth quadrant, the lower part ofthe loop is calculated. For a given coincident-point/saturation-point pair only one quarter of the hysteresis loop is calculatedby (4). By simply changing the relevant sign of the values tochange the quadrant in which they are located, the whole loopmay be calculated accurately.

Pursuit curves may also be useful for predicting the forms ofminor loop excursions, however if the start and end points arenot the coincident and saturation points then the forms of theequations will differ from those given here. There is potentialfor this formalism to be extended to include other modelled hys-teretic behavior, but this would need to be further investigatedin order to validate the limits of the technique.

IV. EXPERIMENTAL VERIFICATION

In order to validate the model an extensive process of exper-imental investigation was performed. A cruciform of annealedmild steel was subjected to a full range of both positive and neg-ative biaxial stress patterns from 120 MPa to 120 MPa, at 40MPa intervals covering every possible combination therein (49stress configurations in all, creating 196 quarter-loops for in-vestigation). The model proved to be robust over the full rangeof both compressive and tensile biaxial stress patterns withinthe elastic limit, and follows the results obtained experimentallyvery closely.

Examples of the results from the zero-stress condition anda representative stress pattern are presented graphically. Fig. 3

Fig. 4. Comparison of the model with experimental data obtained from mild,structural steel with biaxial stresses of 40 MPa perpendicular to and 120 MPaparallel to the applied field. The graph shows the curves calculated for the 2different coincident points to the positive saturation point each with values � �

��� and � � ���� which were estimated using a least squares fit. The fit hasan � � ������.

shows the results of the equation (line) overlaid onto experimen-tally obtained data (points) under zero-loading conditions. Thedata points were obtained as described in previous works [5],[35]–[38]. The data points are from a single acquired loop, notsmoothed or averaged in any way. In Fig. 4, the sample wasunder 40 MPa of stress in the -direction and 120 MPa in the

-direction, the external magnetic field was applied along the-direction. The curves for both increasing and decreasing field

strength are shown. Using Matlab, the exact forms of (5) for thesteel sample used in this study were determined through a leastsquares fitting process to the measured data. Stresses rangedfrom 120 MPa to 120 MPa in both the and directions.Under these conditions the equations for and for a mild steelplate 8 mm thick were fitted to the equations

(6)

(7)

where the coefficients and are material specific. Usingmeasured hysteresis data from a sample of steel under biaxialstrain, it was then possible to fit (4) to the data by altering the

and parameters; then using (6) and (7) it was possible todetermine the stresses experienced by the sample to within 3MPa of that measured by strain gauges, over the 120 MParange investigated.

The equation presented here represents a link between theconceptual model outlined herein at the crystal-unit scale, andactual experimental values as measured under practical condi-tions. The model describes the behavior of the magnetizationvectors of individual ferromagnetic ions in a lattice structureduring the cycling of an externally applied field. This behavior isfurther expanded to include macroscopic, bulk behavior and the

BULTE: PURSUIT OF HYSTERESIS IN POLYCRYSTALLINE FERROMAGNETIC MATERIALS UNDER STRESS 87

effects of externally applied biaxial stresses, parallel and per-pendicular to the applied magnetic field direction.

Through this model the hysteretic behavior of ferromag-netic material is explained, as is the stress dependence ofthis behavior. A lattice binding energy which corresponds tocoincident points used in conjunction with a pursuit curveapproach enables the accurate modelling of ferromagnetic hys-teresis under conditions of biaxially applied external stresses.Using this concept, it should be possible to construct a simple,magnetic measuring device which could determine the absolutestress and strain experienced by a sample of elastically-strainedferromagnetic material with no prior knowledge of the sample’shistory or previous stress/strain conditions; allowing true, quan-titative, nondestructive testing [37].

ACKNOWLEDGMENT

The author would like to thank Dr. R. A. Langman for adviceand assistance in developing this work. Parts of this work areprotected under international patents pending.

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