ductile fracture

8/13/2019 Ductile Fracture

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Ductile Fracture

Ductile FractureOne of the most important and key concepts in the entire fieldof Materials Science and Engineering is fracture. In its simplest form, fracture can bedescribed as a single body being separated into pieces by an imposed stress. Forengineering materials there are only two possible modes of fracture, ductile and brittle. Ingeneral, the main difference between brittle and ductile fracture can be attributed to theamount of plastic deformationthat the material undergoes before fracture occurs. Ductilematerials demonstrate large amounts of plastic deformation while brittle materials show littleor no plastic deformation before fracture. Figure !below", a tensile stress#strain cur$e,represents the degree of plastic deformation e%hibited by both brittle and ductile materialsbefore fracture.

&rack initiation andpropagation are essential to fracture. 'he manner through which the crack propagatesthrough the material gi$es great insight into the mode of fracture. In ductile materials !ductilefracture", the crack mo$es slowly and is accompanied by a large amount of plasticdeformation. 'he crack will usually not e%tend unless an increased stress is applied. On theother hand, in dealing withbrittle fracture, cracks spread $ery rapidly with little or no plasticdeformation. 'he cracks that propagate in a brittle material will continue to grow andincrease in magnitude once they are initiated. (nother important mannerism of crack

propagation is the way in which the ad$ancing crack tra$els through the material. ( crackthat passes through the grains within the material is undergoing transgranular fracture.)owe$er, a crack that propagates along the grain boundaries is termed an intergranularfracture.Figure * !below" shows a scanning electron fractograph of ductile cast iron,e%amining a transgranular fracture surface.
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On bothmacroscopic and microscopic le$els, ductile fracture surfaces ha$e distinct features.Macroscopically, ductile fracture surfaces ha$e larger necking regions and an o$erallrougher appearance than a brittle fracture surface. Figure + !below" shows the macroscopicdifferences between two ductile specimens!a,b" and the brittle specimen !c".

On the microscopic le$el,ductile fracture surfaces also appear rough and irregular. 'he surface consists of manymicro$oids and dimples. Figure !below left" and Figure - !below right" demonstrate themicroscopic ualities of ductile fracture surfaces.


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'he failure of many ductile materials can be attributed to cup and cone fracture. 'his form ofductile fracture occurs in stages that initiate after necking begins. First, small micro$oids formin the interior of the material. /e%t, deformation continues and the micro$oids enlarge to forma crack. 'he crack continues to grow and it spreads laterally towards the edges of thespecimen. Finally, crack propagation is rapid along a surface that makes about a - degreeangle with the tensile stress a%is. 'he new fracture surface has a $ery irregular appearance.'he final shearing of the specimen produces a cup type shape on one fracture surface and acone shape on the ad0acent connecting fracture surface, hence the name, cup and conefracture. Figure 1 !below" shows cup and cone, and brittle fracture in aluminum.

'he &harpyand I2od tests measure the impact energy of a specimen. 3y using an apparatus andimpacting a specimen with a weighted pendulum hammer the impact energy can bemeasured. ( primary use of the &harpy and I2od tests is to determine if a materiale%periences brittle to ductile transition with decreasing temperature. 3rittle to ductiletransition is directly related to the temperature dependency of the impact energy absorbed.

(lso an e%amination of the failure surface can pro$e $ery beneficial. If a section of the failuresurface seems to demonstrate the $isual properties of both brittle and ductile fracture, thenbrittle to ductile transition is e$ident at that temperature range. It is $ery important toremember that with most specimens, there is a fairly wide band of temperatures that supportbrittle to ductile transition. 'herefore, for many specimens it is nearly impossible to predictany one temperature as the transition temperature. In figure 4 !below", a graph is gi$en thatdetermines brittle to ductile transition through an impact test for a 56 hot#rolled steel.


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In most design situations a materialthat demonstrates ductile fracture is usually preferred for se$eral reasons. First andforemost, brittle fracture occurs $ery rapidly and catastrophically without any warning. Ductilematerials plastically deform, thereby slowing the process of fracture and gi$ing ample timefor the problem to be corrected. Second, because of the plastic deformation, more strainenergy is needed to cause ductile fracture. /e%t, ductile materials are considered to be7forgi$ing7 materials, because of their toughness you can make a mistake in the use, designof a ductile material and still the material will probably not fail. (lso, the properties of a ductilematerial can be enhanced through the use of one of the strengthening mechanisms. Strainhardeningis a perfect e%ample, as the ductile material is deformed more and more itsstrength and hardness increase because of the generation of more and more dislocations.'herefore, in engineering applications, especially those that ha$e safety concerns in$ol$ed,

ductile materials are the ob$ious choice. Safety and dependability are the main concerns inmaterial design, but in order to attain these goals there has to be a thorough understandingof fracture, both brittle and ductile. 8nderstanding fracture and failure of materials will leadthe materials engineer to de$elop safer and more dependable materials and products.################################################################################################

Griffith theory
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'he 9riffiths euation describes the relationship between applied nominal stress and crack lengthat fracture, i.e. when it becomes energetically fa$ourable for a crack to grow. 9riffith wasconcerned with the energetics of fracture, and considered the energy changes associated withincremental crack e%tension.For a loaded brittle body undergoing incremental crack e%tension, the only contributors to energychanges are the energy of the new fracture surfaces !two surfaces per crack tip" and the changein potential energy in the body. 'he surface energy term !S" represents energy absorbed in crackgrowth, while the some stored strain energy !U" is released as the crack e%tends !due tounloading of regions ad0acent to the new fracture surfaces". Surface energy has a constant $alueper unit area !or unit length for a unit thickness of body" and is therefore a linear function of !cracklength", while the stored strain energy released in crack growth is a function of !crack length"*,and is hence parabolic. 'hese changes are indicated in the figure below:

'he ne%t step in the de$elopment of 9riffith;s argument was consideration of the rates of energychange with crack e%tension, because the critical condition corresponds to the ma%imum point inthe total energy cur$e, i.e. dW/da< 5, where a< a*. For crack lengths greater than this $alue!under a gi$en applied stress", the body is going to a lower energy state, which is fa$ourable, andhence fast fracture occurs. dW/da< 5 occurs when dS/da< dU/da. 'he sketch below showsthese energy rates, or differentials with respect to a.


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Ris the resistance to crack growth !< dS/da" and Gis the strain energy release rate !< dU/da".=hen fracture occurs, R< Gand we can define Gcrit as the critical value of strain energyrelease, and euate this to R. )ence Gcritrepresents the fracture toughnessof the material. Inplane stressthe 9riffith euation is:

where, to get the fracture stress in M>a !the standard SI engineering unit", the critical strainenergy release rate is in /?m, E is in /?m*, and a is in m. 'his pro$ides an answer in /?m* !>a",which needs to be di$ided by 51 to get the standard engineering unit of M>a. In plane strain:

magnetization or B-H curve
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Magnetization Curves

Any discussion of the magnetic properties of a material is likely to include the type ofgraph known as a magnetization or B-H curve. Various methods are used to produce B-H

curves including one which you can easily replicate.!igure M"A shows how the B-H

curve varies according to the type of material within the field.

#he $curves$ here are all straight lines and have magnetic field strengthas the horizontala%is and the magnetic flu% densityas the vertical a%is. &egative values of H aren$t shown

'ut the graphs are symmetrical a'out the vertical a%is.

!ig. M"A a( is the curve in the a'sence of any material) a vacuum. #he gradient of thecurve is *+.,- which corresponds to the fundamental physical constant /. More on

this later. 0f greater interest is to see how placing a specimen of some material in the

field affects this gradient. Manufacturers of a particular grade of ferrite material usuallyprovide this curve 'ecause the shape reveals how the core material in any component

made from it will respond to changes in applied fiel

1iamagnetic and paramagnetic materials

2magine a hydrogen atom in which a nucleus with a single stationary and positively

charged proton is or'ited 'y a negatively charged electron. Can we view that electron inor'it as a sort of current loop3 #he answer is yes and you might then think that hydrogen

would have a strong magnetic moment.2n fact ordinary hydrogen gas is only very weakly

magnetic. 4ecall that each hydrogen atom is not isolated 'ut is 'onded to one other toform a molecule giving the formula H5 - 'ecause that has a lower chemical energy 6for

H 'y a whopping 5,7 k8 mol-,( than two isolated atoms.

2t is not a coincidence that in these molecules the angular momentum of one electron is

opposite in direction to that of its neigh'our leaving the molecule as a whole with little'y way of magnetic moment. #his 'ehaviour is typical of many su'stances which are

then said to lack a permanent magnetic moment.

9hen a molecule is su':ected to a magnetic field those electrons in or'it planes at a rightangle to the field will change their momentum 6very slightly(. #his is predicted 'y

!araday$s ;awwhich tells us that as the field is increased there will 'e a an induced


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field which the electrons 6'eing charged particles( will e%perience as a force. #his means

that the individual magnetic moments no longer cancel completely and the molecule then

ac=uires an induced magnetic moment. #his 'ehaviour where'y the induced moment isopposite to the applied field is present in all materials and is called diamagnetism.

Hydrogen ammonia 'ismuth copper graphite and other diamagnetic su'stances are

repelled 'y a near'y magnet 6although the effect is e%tremely fee'le(.

#hink of it as a manifestation of ;enz$s law. 1iamagnetic materials are those whose

atoms have only paired electrons.

2n other molecules however such as o%ygen where there are unpaired electrons the

cancellation of magnetic moments 'elonging to the electrons is incomplete. An 05

molecule has a net or permanent magnetic moment even in the a'sence of an e%ternallyapplied field. 2f an e%ternal magnetic field is applied then the electron or'its are still

altered in the same manner as the diamagnets 'ut the permanent moment is usually a

more powerful influence.

#he $poles$ of the molecule tend to line up parallel with the field and reinforce it. >uch

molecules with permanent magnetic moments are called paramagnetic.Although paramagnetic su'stances like o%ygen tin aluminium and copper sulphate are

attracted to a magnet the effect is almost as fee'le as diamagnetism. #he reason is that the

permanent moments are continually knocked out of alignment with the field 'y thermalvi'ration at room temperatures anyway 6li=uid o%ygen at -,7? @C can 'e pulled a'out 'y

a strong magnet(.

"articular materials where the magnetic moment of each atom can 'e made to favour one

direction are said to 'e magnetiza'le.

#he e%tent to which this happens is called themagnetization.!ig. M"A '(a'ove is the

magnetization curvefor diamagnetic materials. 2n diamagnetic su'stances the flu% growsslightly more slowly with the field than it does in a vacuum. #he decrease in gradient is

greatly e%aggerated in the figure - in practice the drop is usually less than one part in

.

!ig. M"A c( is the curve for paramagnetic materials. !lu% growth in this case is again

linear 6at moderate values of H( 'ut slightly faster than in a vacuum. Again the increase

for most su'stances is very slight.

Although neither diamagnetic nor paramagnetic materials are technologically important
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6geophysical surveying is one e%ception( they are much studied 'y physicists and the

terminology of magnetics is enriched there'y. A short uick#ime movie 6?77 B(

demonstrates diamagnetism.

!erromagnetic materials

#he most important class of magnetic materials is the ferromagnets) iron nickel co'altand manganese or their compounds 6and a few more e%otic ones as well(. #he

magnetization curvelooks very different to that of a diamagnetic or paramagnetic

material. 9e might note in passing that although pure manganese is not ferromagnetic the

name of that element shares a common root with magnetism) the Dreek mEgnes lithos -Fstone from MagnesiaF 6now Manisa in #urkey(.

!igure M"B a'ove shows a typical curve for iron. 2t$s important to realize that themagnetization curves for ferromagnetic materials are all strongly dependant upon purity

heat treatment and other factors. However two features of this curve are immediatelyapparent) it really is curved rather than straight 6as with non-ferromagnets( and also thatthe vertical scale is now inteslas6rather than milliteslas as with!igure M"A(.

!igure M"B is a normal magnetization curve 'ecause it starts from an unmagnetizedsample and shows how the flu% densityincreases as the field strengthis increased. Gou

can identify four distinct regions in most such curves.

#hese can 'e e%plained in terms of changes to the material$s magnetic $domains$)

Close to the origin a slow rise due to $reversi'le growth$.

A longer fairly straight stretch representing $irreversi'le growth$.A slower rise representing $rotation$.

An almost flat region corresponding toparamagnetic 'ehaviourand then / - the core

can$t handle any more flu% growth and has saturated.At an atomic level ferromagnetism is e%plained 'y a tendency for neigh'ouring atomic

magnetic momentsto 'ecome locked in parallel with their neigh'ours. #his is only

possi'le at temperatures 'elow the curie point a'ove which thermal disordering causes a

sharp drop inpermea'ilityand degeneration intoparamagnetism.!erromagnetism is
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distinguished from paramagnetism 'y more than :ust permea'ility 'ecause it also has the

important properties of remnanceand coercivity.

!errimagnetic materials

Almost every item of electronic e=uipment produced today contains some ferrimagneticmaterial) loudspeakers motors deflection yokes interference suppressors antenna rods

pro%imity sensors recording heads transformers and inductors are fre=uently 'ased on

ferrites.

#he market is vast.9hat properties make ferrimagnets so u'i=uitous3 #hey possesspermea'ilityto rival

most ferromagnets'ut their eddy current lossesare far lower 'ecause of the material$s

greater electrical resistivity. Also it is practica'le 6if not straightforward( to fa'ricate

different shapes 'y pressing or e%truding - 'oth low cost techni=ues.

9hat is the composition of ferrimagnetic materials3 #hey are in general o%ides of iron

com'ined with one or more of the transition metals such as manganese nickel or zinc

e.g. Mn!e50*. "ermanent ferrimagnets often include 'arium. #he raw material is turnedinto a powder which is then fired in a kiln or sintered to produce a dark gray hard 'rittle

ceramic material having a cu'ic crystalline structure.

At an atomic level the magnetic properties depend upon interaction 'etween the electrons

associated with the metal ions. &eigh'ouring atomic magnetic moments'ecome locked

in anti-parallel with their neigh'ours 6which contrasts with the ferromagnets(. Howeverthe magnetic moments in one direction are weaker than the moments in the opposite

direction leading to an overall magnetic moment.

>aturation

>aturation is a limitation occurring in inductors having a ferromagneticorferrimagneticcore. 2nitially as current is increased theflu%increases in proportion to it 6see figure

M"B(. At some point however further increases in current lead to progressively smaller

increases in flu%.


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proportional to inductance 62 I V J # K ;(. Any drop in inductance therefore causes the

current to rise faster increasing the field strengthand so the core is driven even further

into saturation.

Core manufacturers normally specify the saturation flu% densityfor the particularmaterial used. Gou can also measure saturation using asimple circuit.

#here are two methods 'y which you can calculate flu% if you know the num'er of turnsand either -

#he current the length of the magnetic path and theB-H characteristicsof the material.

#he voltage waveform on a winding and the cross sectional area of the core - see

!araday$s ;aw.

Although saturation is mostly a risk in high power circuits it is still a possi'ility in $small

signal$ applications having many turns on an ungapped core and a 1C 'ias 6such as the

collector current of a transistor(.

2f you find that saturation is likely then you might -

4un the inductor at a lower current

Lse a larger coreAlter the num'er of turns

Lse a core with a lowerpermea'ility

Lse a core with anair gap

or some com'ination thereof - 'ut you$ll need to re-calculate the design in any case.Materials classification

#a'le M"8 categorizes 6in a simplified fashion( each class assigned to a materialaccording to its 'ehaviour in a field.

#a'le M"8) Materials classified 'ytheir magnetic properties.

Class

dependanton B3

1iamagnetic &o

"aramagnetic &o!erromagnetic Ges

Antiferromagnetic Ges

!errimagnetic Ges&i 5

Mn>' 7

Mn0!e50?J ?

G?!e0,5J
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Cr05 ?7

MnAs ?,7

Dd 5N51y 77


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4elative permea'ility

4elative permea'ilityuantity name 4elative

permea'ility

uantity sym'ol /r Lnit sym'ols dimensionless

4elative permea'ility is a very fre=uently used parameter. 2t is a variation upon $straight$

or a'solute permea'ility / 'ut is more useful to you 'ecause it makes clearer how thepresence of a particular material affects the relationship 'etween flu% densityand field

strength. #he term $relative$ arises 'ecause this permea'ility is defined in relation to the

permea'ility of a vacuum /

/r I /K

/

ilicon steel sheet is often made 'y cold rolling to orient the grains along the

laminations6rather than allowing them to lie randomly( giving increased /. 9e call such
http://info.ee.surrey.ac.uk/Workshop/advice/coils/mu/index.html#muhttp://info.ee.surrey.ac.uk/Workshop/advice/coils/terms.html#fdenshttp://info.ee.surrey.ac.uk/Workshop/advice/coils/terms.html#strengthhttp://info.ee.surrey.ac.uk/Workshop/advice/coils/terms.html#strengthhttp://info.ee.surrey.ac.uk/Workshop/advice/coils/mu/index.html#mu0http://info.ee.surrey.ac.uk/Workshop/advice/coils/mu/index.html#muhttp://info.ee.surrey.ac.uk/Workshop/advice/coils/mu/index.html#mu0http://info.ee.surrey.ac.uk/Workshop/advice/coils/mu/index.html#mu0http://info.ee.surrey.ac.uk/Workshop/advice/coils/unit_systems/index.htmlhttp://trimurtuluamie.wetpaint.com/page/magnetization+or+B-H+curve#RANGE!murhttp://info.ee.surrey.ac.uk/Workshop/advice/coils/terms.html#fdenshttp://info.ee.surrey.ac.uk/Workshop/advice/coils/power_loss.html#qhttp://info.ee.surrey.ac.uk/Workshop/advice/coils/power_loss.html#qhttp://info.ee.surrey.ac.uk/Workshop/advice/coils/power_loss.html#eddyhttp://info.ee.surrey.ac.uk/Workshop/advice/coils/mu/index.html#muhttp://info.ee.surrey.ac.uk/Workshop/advice/coils/terms.html#fdenshttp://info.ee.surrey.ac.uk/Workshop/advice/coils/terms.html#strengthhttp://info.ee.surrey.ac.uk/Workshop/advice/coils/terms.html#strengthhttp://info.ee.surrey.ac.uk/Workshop/advice/coils/mu/index.html#mu0http://info.ee.surrey.ac.uk/Workshop/advice/coils/mu/index.html#muhttp://info.ee.surrey.ac.uk/Workshop/advice/coils/mu/index.html#mu0http://info.ee.surrey.ac.uk/Workshop/advice/coils/mu/index.html#mu0http://info.ee.surrey.ac.uk/Workshop/advice/coils/unit_systems/index.htmlhttp://trimurtuluamie.wetpaint.com/page/magnetization+or+B-H+curve#RANGE!murhttp://info.ee.surrey.ac.uk/Workshop/advice/coils/terms.html#fdenshttp://info.ee.surrey.ac.uk/Workshop/advice/coils/power_loss.html#qhttp://info.ee.surrey.ac.uk/Workshop/advice/coils/power_loss.html#eddy


15/23

material anisotropic.

Before you pull any value of / from a data sheet ask yourself if it is appropriate for your

material under the actual conditions under which you use it. !inally if you do not know

the permea'ility of your core then 'uild asimple circuitto measure it.Variant forms of permea'ility and related =uantities

2nitial permea'ility

2nitial permea'ility

uantity name initial permea'ility

uantity sym'ol /i

Lnit sym'ols dimensionless J

2nitial permea'ility descri'es therelative permea'ilityof a material at low values of B

6'elow .,#(. #he ma%imum value for /in a material is fre=uently a factor of 'etween 5and or more a'ove its initial value.

;ow flu% has the advantage that every ferrite can 'e measured at that density without risk

of saturation. #his consistency means that comparison 'etween different ferrites is easy.

Also if you measure the inductance with a normal component 'ridge then you are doingso with respect to the initial permea'ility.

J Although initial permea'ility is usually relative to / you may see /i as ana'solute

permea'ility.


16/23

alias "ermea'ility

of free space

magnetic spaceconstant

magnetic constant

uantity sym'ol /Lnit name henrys per metre

Lnit sym'ols H m-,

Base units kg m s-5 A-5

#hepermea'ilityof a vacuum has a finite value - a'out ,.5R,- Hm-, - and in the >2system 6unlike the cgs system( is denoted 'y the sym'ol /. &ote that this value is

constant withfield strengthand temperature. Contrast this with the situation in

ferromagneticmaterials where / is strongly dependant upon 'oth. Also for practical

purposes most non-ferromagnetic su'stances 6such as wood plastic glass 'one copperaluminium air and water( have a permea'ility almost e=ual to /O that is their relative

permea'ilityis ,..

2n !ig. M"T you see in cross section two long straight conductors spaced one metre

apart in a vacuum. Both carry one ampere. #he field strength due to the current in

conductor A at a distance of one metre may 'e found using AmpUre$s ;aw -

H I 2 K d I , K 65+( A m-,ince we now know 'oth B and H at a distance of , metre from A we calculate the

permea'ility of a vacuum as -
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17/23

/ I B K H I 5R,- K 6, K 65+(( I *+,- Hm-,

2nparamagnetic and diamagneticmaterials the suscepti'ility is given 'y I MK H

where theparamagneticsu'stances have positive suscepti'ilities and the diamagneticsu'stances have negative

suscepti'ilities. #he suscepti'ility of a vacuum is then zero. >uscepti'ility is a strong

contender for the title of $most confusing =uantity in all science$. #here are five reasonsfor this)

#he counterpart topermea'ilityin electrostatics has a distinct name) permittivity.

Lnfortunately the counterpart to suscepti'ility in electrostatics has the same name.

However the electrostatic suscepti'ility should 'e given the sym'ol e. >usceptance hasnothing to do with suscepti'ility.

#here are variant forms of suscepti'ility the main two of which are listed 'elow. Authors

do not always e%plicitly state which variant is 'eing used and worse still there isincomplete agreement a'out the names and sym'ols of each variant. #he sym'ol m is

somewhat overloaded) magnetic suscepti'ility mass suscepti'ility or molar

suscepti'ility3 #ake your pick.Most reference 'ooks and many instruments still present suscepti'ility figures in CD>

units. 0ften the units are not made e%plicit and you are left to deduce them from the

conte%t or the values themselves. #heprocedures for converting to >2are not o'vious.
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18/23

>ome =uite prestigious pu'lications incorrectly a''reviate the units to $per gram$ or $per

kilogram$. )-6

Measurement of suscepti'ility is notoriously difficult. #he slightest whiff ofcontamination 'y iron in the sample will send the e%perimental results off into the

twilight. "u'lished figures fre=uently show differences of SO and S is not rare.

#a'le M">) Magnetic suscepti'ilities

Material v K ,-

Aluminium W5.5Ammonia -,.

Bismuth -,.

Copper -.N5

Hydrogen -.550%ygen W.,N

>ilicon -.?

9ater -.N

#he international sym'ol for suscepti'ility of the ordinary 6$volumetric$( kind is simply

without any su'script. 2>0suggests m to distinguish magnetic suscepti'ility from

electric suscepti'ility 'ut this may risk confusion with mass or molar suscepti'ility. >omewriters have used v to indicate volumetric suscepti'ility. Although electromagnetism is

already up to its ears in su'script soup this seems a good solution.

#a'le M"&) Variant forms ofsuscepti'ility

&ame


19/23

"hysicists like molar suscepti'ility 'ecause their calculations derive from atomic

properties. Deologists like mass suscepti'ility 'ecause they know the weight of their

sample.

#he definition of suscepti'ility given here accords with the >ommerfeld >2 variant. 2n

the ennelly variant has a different definition.

Mass suscepti'ility

Magnetic suscepti'ility 'y mass in the >2

uantity name Magnetic mass

suscepti'ility

alias specificsuscepti'ility

uantity sym'ol X

Lnit sym'ols m? kg-,

Magnetic mass suscepti'ility is simply

X I vK X m? kg-, ommerfeld >2 variant. 2n the

ennelly variant has a different definition.

--------------

#erminology for intrinsic fields within materials -
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20/23

Magnetic moment

Magnetic moment in the >2uantity name magnetic moment

alias magnetic dipole moment

or electromagnetic momentuantity sym'ol m

Lnit name ampere metre s=uared

Lnit sym'ols A m5

#he concept of magnetic moment is the starting point when discussing the 'ehaviour of

magnetic materials within a field. 2f you place a 'ar magnet in a field then it wille%perience a tor=ue or moment tending to align its a%is in the direction of the field. A

compass needle 'ehaves the same. #his tor=ue increases with the strength of the polesand their distance apart. >o the value of magnetic moment tells you in effect $how 'ig amagnet$ you have.

2t is also well known that a current carrying loop in a field also e%periences a tor=ue6electric motors rely on this effect(. Here the tor=ue Y increases with the current i and

the area of the loop A. Z is the angle made 'etween the a%is of the loop normal to its

plane and the field direction.

Y IBR i R A R

sinZ


21/23

"articular materials where the magnetic moment of each atom can 'e made to favour one

direction are said to 'e magnetiza'le. #he e%tent to which this happens is called the

magnetization. Magnetic moment is a vector =uantity which has 'oth direction andmagnitude. #his is important 'ecause although the atoms in most materials may have

magnetic moments they are not easily 'rought into alignment in one direction so the

moments cancel each other leading to weak magnetization.#he


22/23

where m is the total vector sum for the magnetic momentsof all the atoms in a given

volume V 6in m?( of the material. 9e can then say -BI /6HW M( teslas

>ommerfeldfield e=uation

#his e=uation is of theoretical importance 'ecause it highlights a closeness 'etweenH

and M. #he H field is related to $free currents$) for e%ample those flowing from a 'attery

along a piece of wire. M on the other hand is related to the $'ound$ 6$AmpUrian$( currentsof electron or'itals within magnetized materials.

2n practice with ferromagneticmaterials M tends to 'e a very comple% function of H -

including values of H in the past. As a designer of wound components you therefore

pretend instead that B I /H ... and hope for the 'est[

Magnetization occurs not :ust in materials having permanent magnetic moments 'ut also

in any magnetiza'le material in a field which can induce a magnetic moment in itsconstituent atoms. 2n the special case ofparamagnetic and diamagneticmaterials this

magnetization is given 'yM IHA m-,

ommerfeld magnetization. >o

don$t confuse intensity of magnetization withmagnetization

Magnetic polarization

Magnetic polarization in the >2

uantity name Magnetic polarization

alias 2ntensity of
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23/23

magnetization

uantity sym'ol 8

Lnit name teslaLnit sym'ols #

Magnetic polarization is a synonym for intensity of magnetizationin the ennelly variantof the >2
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ductile fracture

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