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Research ArticleThe Magnetic Memory Effect of Ferromagnetic Materials inthe Process of Stress-Magnetism Coupling

GuoqingWang Ping Yan LiwaWei and Zilong Deng

School of Mechanical Engineering Liaoning Shihua University Liaoning Province 113001 China

Correspondence should be addressed to Guoqing Wang wangguoqinglnpueducn

Received 24 April 2017 Accepted 16 July 2017 Published 14 August 2017

Academic Editor Michele Zappalorto

Copyright copy 2017 Guoqing Wang et alThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Ferromagnetic materials can produce the magnetic memory effect under stress This provides a practical method to measurestress concentration The relation between stress and magnetic characteristic is analyzed through energy balance theory Force-magnetism coupling process of Fe-C crystal system is simulated by CASTEP software which is based on first principle Electronband structure electron density of states and atomic magnetic moment in the process of force-magnetism coupling process arecalculated Experimental investigation of the magnetic memory effect of ferromagnetic material under different stresses has beenundertaken in X52 pipeline The results show that the magnetic characteristic of ferromagnetic material weakens under stress andthe magnetic memory signals intensity linearly decreases with the increasing stress When material yields the variation characterof magnetic memory signals suddenly changes and the inflection points of the stress-B curves emerge Experimental investigationis in agreement with the theoretical analysis

1 Introduction

The ferromagnetic materials widely used in engineeringstructure have good mechanical properties and exhibit goodductility and toughness under external stress It can effec-tively resist stress and deformation and ensure the safeoperation of the equipment However the defect or discon-tinuity and other factors often existent in the metal willproduce the stress concentration which induces the locallarge deformation or plastic deformation [1] The plasticdeformation is the early stage of materials damage understress and always becomes the source of materials failureSo it can prevent and reduce the damage accidents throughdetecting the stress concentration and plastic deformationwhich is induced by stress concentration in engineeringfield

The magnetic memory effect is the change of ferro-magnetic materialsrsquo magnetization characteristics under thestress in the weak magnetic field of the earth and whenthe stress is removed the change is irreversible [2] Itis a noncontact nondestructive testing method for stressconcentration and plastic deformation [3ndash5] However thecurrent research on the mechanism of magnetic memory

effect of stress is far from conclusive [6ndash9] The magneticmemory mechanism and signal characteristics are explainedfrom different perspectives by a lot of researches but there isno definite conclusion on them [10ndash14] So the study on themechanism of stress-magnetism coupling of ferromagneticmaterial and the characteristics of magnetic memory effecthas aroused more and more concerns in many countries[15ndash17]

The present paper explores the magnetic memory effectof ferromagnetic materials under stress Firstly the char-acteristics of the magnetic memory signals on the surfaceof the ferromagnetic materials are analyzed from the pointof view of the energy balance of the ferromagnetic bodyThen the process of stress-magnetism coupling is simu-lated by CASTEP software based on the first principleThe influences of stress on the electron band structurethe electron density of states and the atomic magneticmoment are analyzed The quantitative relationship betweenthe stress and the magnetic memory signals is obtained Inthe last part the ferromagnetic pipeline pressure experimentsare carried out to verify the theoretical analysis of themagnetic memory effect of ferromagnetic materials understress

HindawiAdvances in Materials Science and EngineeringVolume 2017 Article ID 1284560 8 pageshttpsdoiorg10115520171284560

2 Advances in Materials Science and Engineering

2 Mechanism Analysis

The basic characteristics of ferromagnetic materials is themagnetic domain structure and magnetic domain is thebasis of all magnetic properties [18] The essence of magneticdomain is an ordered arrangement of atomic magneticmoment in a small region and its formation and existenceare determined by interaction of various energiesThe variousenergies are subject to the law of thermodynamics that thetotal free energy reachesminimumupon system equilibriumHere the magnetic memory signal characteristics whichoccur in the process of force and magnetic coupling areanalyzed through the energy balance theory

The magnetic domain will rotate when the externalmagnetic field 119867 is applied Assume that the angle betweenthe easy magnetization direction of magnetic domain andthe direction of the magnetic field is 1205790 When the magneticmoment is in the direction of easy magnetization the energyof the external magnetic field 119864119867 of ferromagnetic magnet is

119864119867 = minus1205830119872119904119867 cos 1205790 (1)

In the formula 1205830 is the vacuum permeability119872119904 is thesaturation magnetization and 119867 is the external magneticfield strength

When subjected to external magnetic field the magneticmoment will rotate a small angle the energy of the externalmagnetic field 1198641015840119867 of iron magnet is

1198641015840119867 = minus1205830119872119904119867 cos (1205790 minus 120579) (2)

and 1198641015840119867 is less than 119864119867 The greater 120579 is the smaller theexternal magnetic energy is If there are no other obstaclesthe magnetic moment will continue to rotate until 120579 = 1205790But the stress 120590 which exists in a ferromagnetic materialwill produce stress energy 119864120590 which can be expressed in thefollowing formula

119864120590 = 32120582119904120590sin2120579 (3)

In the formula 120582119904 is the magnetostrictive coefficientThe existence of stress can produce the stress anisotropy

which is the same as the magnetic anisotropy which hindersthe rotation of the magnetic moment At this time theinfluence of the magnetization of the material should beconsidered so the total free energy in the unit volume is thesum of the stress energy 119864120590 and the external magnetic fieldenergy 1198641015840119867 that is

119864 = 119864120590 + 1198641015840119867 = 32120582119904120590sin2120579 minus 1205830119872119904119867cos (1205790 minus 120579) (4)

Formula (4) derivation can get relation between 120579 and119867

119889119864119889120579 = 3120582119904120590 sin 120579 cos 120579 minus 1205830119872119904119867 sin (1205790 minus 120579) = 0 (5)

Under weak magnetic field the magnetic moment rota-tion angle is very small so in formula (5) sin 120579 = 120579 cos 120579 asymp 1and sin(1205790 minus 120579) asymp sin 1205790 so formula (5) can be simplified as

120579 = 12058301198721199041198673120582119904120590 sin 1205790 (6)

In addition the magnetization in the direction of themagnetic field can be expressed as

119872 = 119872119904 cos (1205790 minus 120579) (7)

The initial magnetic susceptibility 120594 of the material iscalculated by (6) and (7)

120594 = (119889119872119889119867)119867rarr0

= 119872119904 sin (1205790 minus 120579) 119889120579119889119867asymp 119872119904 sin 1205790 119889120579119889119867 = 120583011987211990423120582119904120590 sin21205790

(8)

Formula (8) is the same direction uniform stress regionof the initial magnetic susceptibility In fact the ferromag-netic material is a multicrystal and the stress directionis different in different grains 1205790 has a variety of valuestherefore it is needed to find out the average value of 120594in different directions Assume that the direction of stressis uniform in the domain and the 1205790 value in formula (8)is averaged and the actual crystal magnetic susceptibility120594(multi) is expressed as

120594(multi) = 120594 = 120583011987211990423120582119904120590 sdot 14120587 int2120587

0int1205870sin21205790 sin 12057901198891205790119889120601

= 120583011987211990423120582119904120590 sdot 23 =2120583011987211990429120582119904120590

(9)

Formula (9) shows that the stress leads to the change ofmagnetic susceptibility and further leads to the change of themagnetization intensity in the local region Under the actionof the external magnetic field119867 the relationship between themagnetic susceptibility and the magnetic flux density 119861 is asfollows

119861 = 1205830 (1 + 120594(multi))119867 = (1205830 + 21205832011987211990429120582119904120590 )119867 (10)

Formula (10) shows that in the areas of stress concen-tration the significantly greater force will reduce the localmagnetic flux density and this leads tomagnetic distortion inthe local region that is the formation of magnetic memorysignal

Here it should be noted that the stress 120590 in formula (10)is equal to external stress 120590ext when ferromagnetic materialsare in the elastic deformation stage While ferromagneticmaterials start to yield a lot of dislocations and plasticdeformation that can reduce the actual stress withinmaterialsappear At present the stress 120590 in formula (10) should bereplaced by equivalent stress 120590eff The relation between 120590extand 120590eff is

120590eff = 119862120590ext (11)

In the formula 119862 is a coefficient which is less than 1 andrelates to the number of dislocations and the degree of plasticdeformation

According to the analysis above the magnetic memorysignals characteristic will change when ferromagnetic mate-rials are in the state of plastic deformation

Advances in Materials Science and Engineering 3

Figure 1 The force-magnetism coupling calculation model

3 Simulation of the Process ofStress-Magnetism Coupling

Based on the theoretical analysis of the magnetic memorysignals characteristics of ferromagnetic material under stressthe first principle method is used to simulate the magneticmemory effect in the process of force magnetic coupling Amodel of the 120572-Fe supercellular crystal with body centeredcubic lattice structure was built by CASTEP module basedon the first principle The lattice constant in the ground stateis set to 28664 A and respectively extended four units inthe 119886 119887 and 119888 three crystallographic directions The 4 times 4times 4 supercell is established In the actual steel the carbonelement is contained in the model and the structure of thecrystalmodel is formed by the replacement of the original cellstructure The crystal calculation model is shown as Figure 1

In the calculation the magnetic memory effect is relatedto the state of the electron spin motion so the electron spinpolarization is added119870 point sampling of the Brillouin zoneis 15 times 15 times 15 and the cut-off energy is calculated as 500 eVThe GGA (generalized gradient approximation) function isused to approximate the calculation process of the force-magnetism coupling The influences of external stress on theelectron density and the gradient distribution of the systemare considered in the GGA function So the more accuratecalculation can be obtained by the selection of GGA function[19]

31 System Energy To simulate the actual water pressurestate the X Y and Z three directionsrsquo tensile stressesare simultaneously applied on the established model Thesystem energy change in the calculation process is shown inFigure 2

The external force will lead to the increase of systemenergy and then the system state becomes unstable Theenergy balance point is achieved through the multiple circu-lation calculation It can be seen fromFigure 2 that the systemenergy undergoes several fluctuations in the early stage andfinally reaches a minimum value and the system reachedthe stable state of energy balance The results show that thecalculation process is convergent

minus1025

minus1020

minus1015

minus1010

Ener

gy (e

v)

minus1005

minus1000

minus995

2 4 6 8 10 12 140Circulation times

Figure 2 System energy

32 Calculation of Electron Band Structure and Density ofStates The magnetic properties of ferromagnetic materialare determined by the atomic magnetic moment whichis determined by the spin movement and distribution ofelectron The crystal structure of ferromagnetic materialswill vary under the action of external stress and then thespin movement and distribution of orbital electrons outsideatomic nucleus are influenced That causes the variationof material magnetic characteristic The variation can beanalyzed through the electron band calculation and densityof states distribution under different stress The calculationresults of electron band structure are shown in Figure 3

It can be seen from Figure 3 that the electron bandstructure of ferromagnetic system in the Brillouin zoneunder different stresses are overlapped and the localizedand itinerant orbital electrons band are distributed near theFermi level (the energy is zero) The electron band is bentand intersects with Fermi level These features indicate thatthe material is ferromagnetic The shapes of the energy bandhave no change under external stress within 0ndash20MPa thatis the material retains ferromagnetic characteristic But theenergy band distributions move far away from the Fermilevel under external stress and then the electron energy nearFermi level gradually decreases with the increasing stresswhich will cause the variation of ferromagnetic characteristicof material

The electron energy band distribution is related to thetotal electrons density of states which are the difference ofdensity of states between spin up electrons and spin downelectrons In order to further analyze the influence of thestress on ferromagnetic characteristics the total electrondensity states are calculated under different stresses Thecalculation results are shown in Figure 4

It can be seen from Figure 4 that near the Fermi levelthe peak values and valley values of the total electron densityof states distribution curve significantly decrease under theexternal stress which indicate that the numbers of the spinelectron near the Fermi level gradually decrease and theferromagnetic characteristic of material gradually weakenswith the increasing external stress


minus8

minus6

minus4

minus2

0

2

4

6En

ergy

(ev)

F GZQGBrillouin zone(a)

minus8

minus6

minus4

minus2

0

2

4

6

Ener

gy (e

v)

F GZQGBrillouin zone(b)

minus8

minus6

minus4

minus2

0

2

4

6

Ener

gy (e

v)

F GZQGBrillouin zone(c)

minus8

minus6

minus4

minus2

0

2

4

6

Ener

gy (e

v)

F GZQGBrillouin zone(d)

minus8

minus6

minus4

minus2

0

2

4

6

Ener

gy (e

v)

F GZQGBrillouin zone(e)

Figure 3 Electron band structure (a) under 0MPa stress (b) under 5MPa stress (c) under 10MPa stress (d) under 15MPa stress (e) under20MPa stress

33 Calculation of Atomic Magnetic Moment The variationfeatures of the magnetic characteristics of ferromagneticmaterials under different external stresses have been quali-tatively analyzed by calculating the electron band structureand density of states In order to qualitatively analyze the

magnetic memory effect of ferromagnetic materials in theprocess of stress-magnetism coupling the atomic magneticmoments of ferromagnetic materials under the externalstresses within 0ndash20MPa are calculated and the results areshown in Figure 5


00

05

10

15

20

25

30

Tota

l den

sity

of st

ate (

elec

trons

ev)

0 5 10minus5minus10Energy (ev)

(a)

00

05

10

15

20

25

30

Tota

l den

sity

of st

ate (

elec

trons

ev)


(b)

00

05

10

15

20

25

30

Tota

l den

sity

of st

ate (

elec

trons

ev)


(c)

00

05

10

15

20

25

30To

tal d

ensit

y of

stat

e (el

ectro

nse

v)


(d)

00

05

10

15

20

25

30

Tota

l den

sity

of st

ate (

elec

trons

ev)


(e)

Figure 4 Electron density of states (a) under 0MPa stress (b) under 5MPa stress (c) under 10MPa stress (d) under 15MPa stress (e) under20MPa stress


5 10 15 200Stress (MPa)

21708

21710

21712

21714

21716

21718

21720

21722

Atom

ic m

agne

tic m

omen

t (u B

ato

m)

Figure 5The relation between atomicmagneticmoment and stress

It can be seen from Figure 5 that the atomic magneticmoment linearly decreases with the increasing external stressat early stage and there is an inflection point in the curvewhen the stress increases to 15MPaThe curve slope decreasesunder greater stress than 15MPa

The magnetization of materials is related to the atomicmagneticmomentThe relationship between the atomicmag-netic moment and magnetization intensity119872 is as follows

119872 = sum120583119895119881 (12)

In the formula 120583119895 is the atomic magnetic moment Vis the solid volume and sum120583119895 is the sum of all the atomicmagnetic moments in solid

Thus the relationship between 120583119895 and the magnetic fluxdensity 119861 is

119861 = 1205830 (119867 +119872) = 1205830 (119867 + sum120583119895119881 ) (13)

According to the calculations and analyses above theconclusion can be drawn that the magnetic flux densityof ferromagnetic materials varies and there exists suddenchange in the process of stress-magnetism coupling

4 Experimental Section

41 Specimen Preparation In order to study the magneticmemory signal characters of ferromagnetic materials in theprocess of stress-magnetism coupling the pipeline pressureexperiment is designed The experiment material is ferro-magnetic X80 steel which is applied widely in engineeringThe external diameter of pipeline is 273mm and the wallthickness is 6mm To eliminate the influence of the edgeeffect produced by specimens with finite dimensions on weakmagnetic signal measurement the length of experimentalpipeline is 120m Signal measurement is carried out on themiddle pipeline with the length range of 15mm The straingages monitored the stress-strain state of pipeline under

different stress which are pasted on positions 1 and 2 whichare shown in Figure 6 Weak magnetic signals sensors whichare based on fluxgate and used to measure the magneticfield intensity in 119883 direction (normal direction) and Ydirection (axial direction) are fixed beside the strain gagesThe measurement range of sensors is 0ndash100000 nT and themeasurement sensitivity is 1 nT

42 Experimental Procedure The two ends of the pipelinewere sealed by high pressure blind plateThere were the pres-sure sensor which was used to monitor the pressure withinthe pipeline and thewater injection gatewhichwas connectedto the pressure deviceThe strain changesweremonitored andrecorded during the process of pressure increases from zeroThe pressure device stopped when inflection point of stress-strain curve emerged (ie the plastic deformation of pipelineemerged) The dates of stress-strain and magnetic memorysignals were recorded in the whole processThen the pressurewithin the pipeline drops to zero To verify the reliability ofthe data the experimental procedure was repeated two times

43 Results and Discussion In the three experimental pro-cedures the variation characteristics of stress-strain andmagneticmemory signals in position 1 were the same as thosein position 2 So here the dates only in position 1 are used toanalyze the experimental resultsThe stress-strain diagram inthree experimental procedures was shown in Figure 7

It can be seen from Figure 7 that the strain of thepipeline linearly increases with increasing stress in the elasticdeformation stage and the relation between stress and strainis proportional When the stress increases to 15MPa theinflection point of curve emerges in the first experimentalprocedure which indicates that the materials start to yieldand pipeline start to produce plastic deformationThe stressescorresponding to the inflection points of stress-strain curvewhich show materials yield are respectively 155MPa and16MPa in the second and the third experimental procedureThe major reason for the increase of yield stress is that theplastic deformation causes the materials to hardenThemag-netic memory signals testing results in three experimentalprocedures were shown in Figure 8

It can be seen from Figure 8 that in three experimentalprocedures the magnetic flux density 119861 both in X directionand in Y direction linearly decreases with increasing stressand the inflection points of the stress-119861 curves emerge andthe variation character of magnetic memory signals suddenlychanges when the pipeline yields In the plastic deformationstage the slope of all the curves decreases and the variationrate of magnetic induction intensity decreases

5 Conclusion

In the here presented paper systematic investigation ofmagnetic memory effect of ferromagnetic material understress has been performed via theory and experimentalapproaches The primary cause of magnetic memory effectformation is that the stress causes the variation of themagnetic susceptibility and atomic magnetic moment offerromagnetic material The variety features of magnetic


2

Strain gage

Pressure sensorWater injection gate

1

Measurement sensor

(a) (b)

Figure 6 Experimental device (a) the schematic diagram of experimental device (b) The photograph in the scene

2 4 6 8 10 12 14 16 180Stress (MPa)

FirstSecond Third

0

500

1000

1500

2000

2500

Stra

in ()

Figure 7 The stress-strain curve in three experimental procedures

Feature points

FirstSecond Third

2 4 6 8 10 12 14 16 180Stress (MPa)

21000

22000

23000

24000

25000

26000

27000

B (n

T)

(a)

Feature points

FirstSecond Third

33500

34000

34500

35000B (n

T)

35500

36000

36500

37000

2 4 6 8 10 12 14 16 180Stress (MPa)

(b)

Figure 8 The signals character of magnetic memory in three experimental procedures (a) The 119883 direction (b) The 119884 direction


characteristic are obtained by calculating the electron bandstructure electron density of states and atomic magneticmoment It is significant that there is a linear relationshipbetween stress and magnetic memory signals and the stress-B curves have the inflection points which indicate that thematerials yield and start to produce plastic deformation Thetheory investigation is validated by the pressure experimentThis paper provides a convenient and practical approach todetecting the stress concentration and plastic deformationand predicting the damage of ferromagnetic material

Disclosure

The founding sponsors had no role in the design of the studyin the collection analyses or interpretation of data in thewriting of the manuscript and in the decision to publish theresults

Conflicts of Interest

The authors declare no conflicts of interest

Acknowledgments

This work was funded by project supported by the NationalNatural Science Foundation of China (Grant no 61571308)and project supported by Liaoning Education Department ofChina (Grant no L2016009)

References

[1] S Yoichi ldquoStress Concentration Problemsrdquo Mathematical andComputational Analyses of Cracking Formation vol 2 pp 17ndash30 2014

[2] A Dubov A Dubov and S Kolokolnikov ldquoApplication of themetal magnetic memory method for detection of defects at theinitial stage of their development for prevention of failures ofpower engineering welded steel structures and steam turbinepartsrdquoWelding in the World vol 58 no 2 pp 225ndash236 2014

[3] S M Kolokolnikov A A Dubov and A Y MarchenkovldquoDetermination of mechanical properties of metal of weldedjoints by strength parameters in the stress concentration zonesdetected by the metal magnetic memory methodrdquo Welding inthe World vol 58 no 5 pp 699ndash706 2014

[4] R Maciej ldquoMetal magnetic memory testing of welded joints offerritic and austenitic steelsrdquoNDTampE International vol 44 pp305ndash310 2011

[5] A Dubov and S Kolokolnikov ldquoThe metal magnetic memorymethod application for online monitoring of damage develop-ment in steel pipes and welded joints specimensrdquoWelding in theWorld vol 57 no 1 pp 123ndash136 2013

[6] C SaraMAugusto andUMariangela ldquoModelingUltrasoundsfor Nondestructive Testing Applicationsrdquo in Ultrasonic Nonde-structive Evaluation Systems vol 12 pp 47ndash82 2014

[7] Sorabh A Gupta and K Chandrasekaran ldquoFinite ElementModeling ofMagnetic Flux Leakage fromMetal Loss Defects inSteel Pipelinerdquo Journal of Failure Analysis and Prevention vol16 no 2 pp 316ndash323 2016

[8] J Jongwoo and L Jinyi ldquoNon-destructive evaluation of cracks ina paramagnetic specimen with low conductivity by penetration

of magnetic fluidrdquo NDT amp E International vol 42 pp 297ndash3032009

[9] K Yao Z D Wang B Deng and K Shen ldquoExperimentalResearch on Metal Magnetic Memory Methodrdquo ExperimentalMechanics vol 52 no 3 pp 305ndash314 2012

[10] B Liu W Sun and Y Lin ldquoThe Study of ElectromagneticStress Testing Method on Oil-Gas Pipelines Based on WTrdquoGeomaterials vol 04 pp 55ndash63 2014

[11] H Hauser Y Melikhov and D C Jiles ldquoExamination of theequivalence of ferromagnetic hysteresis models describing thedependence of magnetization on magnetic field and stressrdquoIEEE Transactions on Magnetics vol 45 no 4 pp 1940ndash19492009

[12] B Liu Y Y He andH Zhang ldquoStudy on characteristics of mag-netic memory testing signal based on the stress concentrationfieldrdquo IET Science Measurement amp Technology vol 11 pp 2ndash82017

[13] M Roskosz and P Gawrilenko ldquoAnalysis of changes in residualmagnetic field in loaded notched samplesrdquo NDT amp E Interna-tional vol 41 pp 570ndash576 2008

[14] J Leng Y Liu G Zhou and Y Gao ldquoMetal magnetic memorysignal response to plastic deformation of low carbon steelrdquoNDTamp E International vol 55 pp 42ndash46 2013

[15] L J Yang B Liu and L J Chen ldquoThequantitative interpretationbymeasurement using themagneticmemorymethod (MMM)-based on density functional theoryrdquoNDTampE International vol55 pp 15ndash20 2013

[16] J L Ren C Chen C K Liu et al ldquoExperimental research onmicrocosmic mechanism of stress-magnetic effect for magneticmemory testingrdquo Journal of Aeronautical Materials vol 28 pp41ndash44 2008

[17] L Dong B Xu S Dong et al ldquoStress dependence of thespontaneous stray field signals of ferromagnetic steelrdquoNDT andE International vol 42 no 4 pp 323ndash327 2009

[18] A Lubk S Gemming andNA Spaldin ldquoFirst-principles studyof ferroelectric domain walls in multiferroic bismuth ferriterdquoPhysical ReviewB -CondensedMatter andMaterials Physics vol80 no 10 Article ID 104110 2009

[19] Y KAkehashi and M Patoary ldquoFirst-Principles DynamicalCoherent-Potential Approximation Approach to the Ferromag-netism of Fe Co andNirdquo Journal of the Physical Society of Japanvol 11 pp 1683ndash1694 2011

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2 Mechanism Analysis

The basic characteristics of ferromagnetic materials is themagnetic domain structure and magnetic domain is thebasis of all magnetic properties [18] The essence of magneticdomain is an ordered arrangement of atomic magneticmoment in a small region and its formation and existenceare determined by interaction of various energiesThe variousenergies are subject to the law of thermodynamics that thetotal free energy reachesminimumupon system equilibriumHere the magnetic memory signal characteristics whichoccur in the process of force and magnetic coupling areanalyzed through the energy balance theory

The magnetic domain will rotate when the externalmagnetic field 119867 is applied Assume that the angle betweenthe easy magnetization direction of magnetic domain andthe direction of the magnetic field is 1205790 When the magneticmoment is in the direction of easy magnetization the energyof the external magnetic field 119864119867 of ferromagnetic magnet is

119864119867 = minus1205830119872119904119867 cos 1205790 (1)

In the formula 1205830 is the vacuum permeability119872119904 is thesaturation magnetization and 119867 is the external magneticfield strength

When subjected to external magnetic field the magneticmoment will rotate a small angle the energy of the externalmagnetic field 1198641015840119867 of iron magnet is

1198641015840119867 = minus1205830119872119904119867 cos (1205790 minus 120579) (2)

and 1198641015840119867 is less than 119864119867 The greater 120579 is the smaller theexternal magnetic energy is If there are no other obstaclesthe magnetic moment will continue to rotate until 120579 = 1205790But the stress 120590 which exists in a ferromagnetic materialwill produce stress energy 119864120590 which can be expressed in thefollowing formula

119864120590 = 32120582119904120590sin2120579 (3)

In the formula 120582119904 is the magnetostrictive coefficientThe existence of stress can produce the stress anisotropy

which is the same as the magnetic anisotropy which hindersthe rotation of the magnetic moment At this time theinfluence of the magnetization of the material should beconsidered so the total free energy in the unit volume is thesum of the stress energy 119864120590 and the external magnetic fieldenergy 1198641015840119867 that is

119864 = 119864120590 + 1198641015840119867 = 32120582119904120590sin2120579 minus 1205830119872119904119867cos (1205790 minus 120579) (4)

Formula (4) derivation can get relation between 120579 and119867

119889119864119889120579 = 3120582119904120590 sin 120579 cos 120579 minus 1205830119872119904119867 sin (1205790 minus 120579) = 0 (5)

Under weak magnetic field the magnetic moment rota-tion angle is very small so in formula (5) sin 120579 = 120579 cos 120579 asymp 1and sin(1205790 minus 120579) asymp sin 1205790 so formula (5) can be simplified as

120579 = 12058301198721199041198673120582119904120590 sin 1205790 (6)

In addition the magnetization in the direction of themagnetic field can be expressed as

119872 = 119872119904 cos (1205790 minus 120579) (7)

The initial magnetic susceptibility 120594 of the material iscalculated by (6) and (7)

120594 = (119889119872119889119867)119867rarr0

= 119872119904 sin (1205790 minus 120579) 119889120579119889119867asymp 119872119904 sin 1205790 119889120579119889119867 = 120583011987211990423120582119904120590 sin21205790

(8)

Formula (8) is the same direction uniform stress regionof the initial magnetic susceptibility In fact the ferromag-netic material is a multicrystal and the stress directionis different in different grains 1205790 has a variety of valuestherefore it is needed to find out the average value of 120594in different directions Assume that the direction of stressis uniform in the domain and the 1205790 value in formula (8)is averaged and the actual crystal magnetic susceptibility120594(multi) is expressed as

120594(multi) = 120594 = 120583011987211990423120582119904120590 sdot 14120587 int2120587

0int1205870sin21205790 sin 12057901198891205790119889120601

= 120583011987211990423120582119904120590 sdot 23 =2120583011987211990429120582119904120590

(9)

Formula (9) shows that the stress leads to the change ofmagnetic susceptibility and further leads to the change of themagnetization intensity in the local region Under the actionof the external magnetic field119867 the relationship between themagnetic susceptibility and the magnetic flux density 119861 is asfollows

119861 = 1205830 (1 + 120594(multi))119867 = (1205830 + 21205832011987211990429120582119904120590 )119867 (10)

Formula (10) shows that in the areas of stress concen-tration the significantly greater force will reduce the localmagnetic flux density and this leads tomagnetic distortion inthe local region that is the formation of magnetic memorysignal

Here it should be noted that the stress 120590 in formula (10)is equal to external stress 120590ext when ferromagnetic materialsare in the elastic deformation stage While ferromagneticmaterials start to yield a lot of dislocations and plasticdeformation that can reduce the actual stress withinmaterialsappear At present the stress 120590 in formula (10) should bereplaced by equivalent stress 120590eff The relation between 120590extand 120590eff is

120590eff = 119862120590ext (11)

In the formula 119862 is a coefficient which is less than 1 andrelates to the number of dislocations and the degree of plasticdeformation

According to the analysis above the magnetic memorysignals characteristic will change when ferromagnetic mate-rials are in the state of plastic deformation








minus1025

minus1020

minus1015

minus1010

Ener

gy (e

v)

minus1005

minus1000

minus995








minus8

minus6

minus4

minus2

0

2

4

6En

ergy

(ev)


minus8

minus6

minus4

minus2

0

2

4

6

Ener

gy (e

v)


minus8

minus6

minus4

minus2

0

2

4

6

Ener

gy (e

v)


minus8

minus6

minus4

minus2

0

2

4

6

Ener

gy (e

v)


minus8

minus6

minus4

minus2

0

2

4

6

Ener

gy (e

v)






00

05

10

15

20

25

30

Tota

l den

sity

of st

ate (

elec

trons

ev)


(a)

00

05

10

15

20

25

30

Tota

l den

sity

of st

ate (

elec

trons

ev)


(b)

00

05

10

15

20

25

30

Tota

l den

sity

of st

ate (

elec

trons

ev)


(c)

00

05

10

15

20

25

30To

tal d

ensit

y of

stat

e (el

ectro

nse

v)


(d)

00

05

10

15

20

25

30

Tota

l den

sity

of st

ate (

elec

trons

ev)


(e)



5 10 15 200Stress (MPa)

21708

21710

21712

21714

21716

21718

21720

21722

Atom

ic m

agne

tic m

omen

t (u B

ato

m)




119872 = sum120583119895119881 (12)



119861 = 1205830 (119867 +119872) = 1205830 (119867 + sum120583119895119881 ) (13)









5 Conclusion



2

Strain gage


1

Measurement sensor

(a) (b)


2 4 6 8 10 12 14 16 180Stress (MPa)

FirstSecond Third

0

500

1000

1500

2000

2500

Stra

in ()


Feature points

FirstSecond Third

2 4 6 8 10 12 14 16 180Stress (MPa)

21000

22000

23000

24000

25000

26000

27000

B (n

T)

(a)

Feature points

FirstSecond Third

33500

34000

34500

35000B (n

T)

35500

36000

36500

37000

2 4 6 8 10 12 14 16 180Stress (MPa)

(b)




Disclosure




Acknowledgments


References




























CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of









minus1025

minus1020

minus1015

minus1010

Ener

gy (e

v)

minus1005

minus1000

minus995








minus8

minus6

minus4

minus2

0

2

4

6En

ergy

(ev)


minus8

minus6

minus4

minus2

0

2

4

6

Ener

gy (e

v)


minus8

minus6

minus4

minus2

0

2

4

6

Ener

gy (e

v)


minus8

minus6

minus4

minus2

0

2

4

6

Ener

gy (e

v)


minus8

minus6

minus4

minus2

0

2

4

6

Ener

gy (e

v)






00

05

10

15

20

25

30

Tota

l den

sity

of st

ate (

elec

trons

ev)


(a)

00

05

10

15

20

25

30

Tota

l den

sity

of st

ate (

elec

trons

ev)


(b)

00

05

10

15

20

25

30

Tota

l den

sity

of st

ate (

elec

trons

ev)


(c)

00

05

10

15

20

25

30To

tal d

ensit

y of

stat

e (el

ectro

nse

v)


(d)

00

05

10

15

20

25

30

Tota

l den

sity

of st

ate (

elec

trons

ev)


(e)



5 10 15 200Stress (MPa)

21708

21710

21712

21714

21716

21718

21720

21722

Atom

ic m

agne

tic m

omen

t (u B

ato

m)




119872 = sum120583119895119881 (12)



119861 = 1205830 (119867 +119872) = 1205830 (119867 + sum120583119895119881 ) (13)









5 Conclusion



2

Strain gage


1

Measurement sensor

(a) (b)


2 4 6 8 10 12 14 16 180Stress (MPa)

FirstSecond Third

0

500

1000

1500

2000

2500

Stra

in ()


Feature points

FirstSecond Third

2 4 6 8 10 12 14 16 180Stress (MPa)

21000

22000

23000

24000

25000

26000

27000

B (n

T)

(a)

Feature points

FirstSecond Third

33500

34000

34500

35000B (n

T)

35500

36000

36500

37000

2 4 6 8 10 12 14 16 180Stress (MPa)

(b)




Disclosure




Acknowledgments


References




























CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of



minus8

minus6

minus4

minus2

0

2

4

6En

ergy

(ev)


minus8

minus6

minus4

minus2

0

2

4

6

Ener

gy (e

v)


minus8

minus6

minus4

minus2

0

2

4

6

Ener

gy (e

v)


minus8

minus6

minus4

minus2

0

2

4

6

Ener

gy (e

v)


minus8

minus6

minus4

minus2

0

2

4

6

Ener

gy (e

v)






00

05

10

15

20

25

30

Tota

l den

sity

of st

ate (

elec

trons

ev)


(a)

00

05

10

15

20

25

30

Tota

l den

sity

of st

ate (

elec

trons

ev)


(b)

00

05

10

15

20

25

30

Tota

l den

sity

of st

ate (

elec

trons

ev)


(c)

00

05

10

15

20

25

30To

tal d

ensit

y of

stat

e (el

ectro

nse

v)


(d)

00

05

10

15

20

25

30

Tota

l den

sity

of st

ate (

elec

trons

ev)


(e)



5 10 15 200Stress (MPa)

21708

21710

21712

21714

21716

21718

21720

21722

Atom

ic m

agne

tic m

omen

t (u B

ato

m)




119872 = sum120583119895119881 (12)



119861 = 1205830 (119867 +119872) = 1205830 (119867 + sum120583119895119881 ) (13)









5 Conclusion



2

Strain gage


1

Measurement sensor

(a) (b)


2 4 6 8 10 12 14 16 180Stress (MPa)

FirstSecond Third

0

500

1000

1500

2000

2500

Stra

in ()


Feature points

FirstSecond Third

2 4 6 8 10 12 14 16 180Stress (MPa)

21000

22000

23000

24000

25000

26000

27000

B (n

T)

(a)

Feature points

FirstSecond Third

33500

34000

34500

35000B (n

T)

35500

36000

36500

37000

2 4 6 8 10 12 14 16 180Stress (MPa)

(b)




Disclosure




Acknowledgments


References




























CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of



00

05

10

15

20

25

30

Tota

l den

sity

of st

ate (

elec

trons

ev)


(a)

00

05

10

15

20

25

30

Tota

l den

sity

of st

ate (

elec

trons

ev)


(b)

00

05

10

15

20

25

30

Tota

l den

sity

of st

ate (

elec

trons

ev)


(c)

00

05

10

15

20

25

30To

tal d

ensit

y of

stat

e (el

ectro

nse

v)


(d)

00

05

10

15

20

25

30

Tota

l den

sity

of st

ate (

elec

trons

ev)


(e)



5 10 15 200Stress (MPa)

21708

21710

21712

21714

21716

21718

21720

21722

Atom

ic m

agne

tic m

omen

t (u B

ato

m)




119872 = sum120583119895119881 (12)



119861 = 1205830 (119867 +119872) = 1205830 (119867 + sum120583119895119881 ) (13)









5 Conclusion



2

Strain gage


1

Measurement sensor

(a) (b)


2 4 6 8 10 12 14 16 180Stress (MPa)

FirstSecond Third

0

500

1000

1500

2000

2500

Stra

in ()


Feature points

FirstSecond Third

2 4 6 8 10 12 14 16 180Stress (MPa)

21000

22000

23000

24000

25000

26000

27000

B (n

T)

(a)

Feature points

FirstSecond Third

33500

34000

34500

35000B (n

T)

35500

36000

36500

37000

2 4 6 8 10 12 14 16 180Stress (MPa)

(b)




Disclosure




Acknowledgments


References




























CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of



5 10 15 200Stress (MPa)

21708

21710

21712

21714

21716

21718

21720

21722

Atom

ic m

agne

tic m

omen

t (u B

ato

m)




119872 = sum120583119895119881 (12)



119861 = 1205830 (119867 +119872) = 1205830 (119867 + sum120583119895119881 ) (13)









5 Conclusion



2

Strain gage


1

Measurement sensor

(a) (b)


2 4 6 8 10 12 14 16 180Stress (MPa)

FirstSecond Third

0

500

1000

1500

2000

2500

Stra

in ()


Feature points

FirstSecond Third

2 4 6 8 10 12 14 16 180Stress (MPa)

21000

22000

23000

24000

25000

26000

27000

B (n

T)

(a)

Feature points

FirstSecond Third

33500

34000

34500

35000B (n

T)

35500

36000

36500

37000

2 4 6 8 10 12 14 16 180Stress (MPa)

(b)




Disclosure




Acknowledgments


References




























CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of



2

Strain gage


1

Measurement sensor

(a) (b)


2 4 6 8 10 12 14 16 180Stress (MPa)

FirstSecond Third

0

500

1000

1500

2000

2500

Stra

in ()


Feature points

FirstSecond Third

2 4 6 8 10 12 14 16 180Stress (MPa)

21000

22000

23000

24000

25000

26000

27000

B (n

T)

(a)

Feature points

FirstSecond Third

33500

34000

34500

35000B (n

T)

35500

36000

36500

37000

2 4 6 8 10 12 14 16 180Stress (MPa)

(b)




Disclosure




Acknowledgments


References




























CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of




Disclosure




Acknowledgments


References




























CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of









CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of
