UDC 539.374.1 Determination of integral …...Determination of integral characteristics of stress state of the point during plastic deformation in conditions of volume loading UDC

153Metallurgical and Mining IndustryNo.11 — 2015

Materials science3. Laber K., Dyja H., KałamorzM.: „Analysis

ofthetechnologyofrolling5,5mm-diameterwirerodofcoldupsettingsteelintheMorganBlock mill”, Metallurgija, vol. 54, issue: 2,Apr-Jun2015,pp.415-418;

4. Laber K., Dyja H., Koczurkiewicz B., Sa-wicki S.: „Fizyczne modelowanie procesu

walcowania walcówki ze stali 20MnB4”,VIKonferencja Naukowa WALCOWNICTWO2014.Procesy-Narzędzia–Materiały,20.10-22.10.2014r.,Ustroń,s.37-42;

5. PN-68/H-04500 „Badania dylatometrycznemetaliiichstopów”

Determination of integral characteristics of stress state of the point during plastic deformation in conditions of

volume loading

UDC539.374.1

Chigirinsky V.V.

D.Sc. in engineering, professor,Head of ZNTU Metal Forming Department

Zaporizhzhya National Technical University,

Zaporozhye, UkraineE-mail: [email protected]

Lenok A. A.

Assistant of ZNTU Metal Forming Department

Zaporozhye National Technical University,Zaporozhye, Ukraine

E-mail: [email protected]

Echin S. M.

Assistant of ZNTU Metal Forming Department

Zaporozhye National Technical University,Zaporozhye, Ukraine


mailto:[email protected]



Metallurgical and Mining Industry154 No.11 — 2015

Materials scienceAbstractItwassetandsolvedtheclosedproblemofthetheoryofplasticityintermsofvolumeloading.Circularityofsolutiondoesnotcomplicatethetask,butsimplifiesassixequationsofkinematicpartofataskgivestothesolutionatachoiceofcombinationsofflatfunctions.Accordingtotheequationsofconnection,suchrequirementsofdeformationsareimposedtothestaticpartofataskatdeterminationofstress.Various solutions for themaking components of a tensor of stresswere obtained.Dependencies forintegralcharacteristicsofstressstateofdeformationpointwereshown.Thekernelofthesolutionofintensityofnormalstress,whichcharacterizesthegeneralizedpointstressparameterswasdefined.Key words: INTEGRATED CHARACTERISTICS, PLASTIC MEDIUM, STRESS STATE,INTENSITY,VOLUMELOADING

IntroductionAtthesolutionofpracticalandanumberofthe-

oreticaltasksthereisaneedtodefinetheintegratedcharacteristicsoftensionofapoint.Amongtheseare

theintensitiesoftangentialnormalstress iiT σ, .Attheknownvaluesofcomponentsofstresstensor,in-tensitiesaredefinedbythefollowingexpressions

,(1)

Further theexpressions (1)canbeusedfordefi-nitionofmechanicalcharacteristicsofconcretesteelgradeinconditionsofhotorcoldprocessingineachpointofthedeformationzone.

Solution of space problem in analytical formcausesgreatmathematicaltroubles.Thatiswhytherealways appear questions connectedwith simplifica-tionsinstatementanditssolution[1].Closedtaskofthe theoryofplasticityconnectedwithdefinitionoffieldsof tension anddeformations, their correlationbetweeneachother, satisfactionofboundarycondi-tions,complicatesconsiderablythesolution.Itresultsininsuperabledifficultiesinthesolution.However,itispossibletostopontheoptionswhenclosedprob-lemdefinitiondoesnot complicate,but simplifiesatask.Thesystemof thedifferentialequationsofki-

nematic part of a task canbe simplified if to use acombination of flat functions, and then through theequationsofconnectiontocometostaticpartwhendeterminingatensionofapoint.Atsuchschemeofsolution,itispossibletoconsidertheoptionsatisfy-ingallsystemoftheequationsofthetheoryofplas-ticityintheclosedview.Thus,thestaticpartofataskisprovedandsupportedbythekinematic.Besides,atthesolutionofpracticaltasksitisnotalwayspossibletofindreliableresultforanalyticaldeterminationoftensionanddeformationsintransitionzonesfromonepartofdeformationzoneintotheadjacent.

Thefigure1showsstresstensorcomponentsandcontact surface of deformation zone in plan takingintoaccounttransitionareasfromonezoneofplasticflowintoanother.

Figure 1. Stresstensorcomponentsandmechanismoftangentialstressonthecontact


Materials scienceAt the transition areas tangential stresses τxz and

τyzchange along the value and direction. On theboundarylineofmetalflow,axesY ,Y , xyτ ,tangen-tialstressesτxy,τyz, τzxareequaltozero[2].Thisallowstosupposethatalongtheseaxesthereisplane-straincondition.

Problem statement Taking into account the last notes, there consid-

eredthefollowingstatementofclosedvolumetaskofplastictheoryincludingtheequations[3]...[7]

1.Equilibriumequations

(1)

2.Generalizedequilibriumequations

, ,(2)

.

3.Equationsofconnection

,

(3)

4.Compatibilityequationsofstrainrate

(4а)

.(4b)

5.Boundaryconditions

( )11111 2ατ −Α⋅= FSinkn ,

( )22222 2ατ −Α⋅= FSinkn ,(5)( )22222 22 αβγ −Β⋅⋅= FSinn ,

( )22222 22 αβγ −Β⋅⋅= FSinn .

In result of arrangement we have 19 unknownvariables (

) and 19equationsincluding4equationsforboundarycondi-tions(1)…(5),where 321 ,, kkk -aretheresistancetoflowshear inplains ; 321 ,, βββ -arethe intensities of shear speed of deformation in thesameplains.Thetaskisstaticallydeterminable.

Structure of solution Useofflatfunctionsorcombinationsofflatfunc-

tions in the solution isdefinedbyobjective to sim-plifyaspaceproblemandfullysatisfythesystemofequationsof plasticity theory (1)...(5).Thegeneral-ized equationsofbalance (2), at knowndifferencesof normal tension, givemathematical possibility tocharacterizetransitionareasbetweenzonesofplasticcurrent.Theanalysisshowsthatthereareoptions,atwhichtheirapplicationcomestrueatthesolutionof


Materials sciencespacetasksofplasticitytheory.

Intheworks[4]…[7]analyticalsolutionofplanetaskofplasticitytheorywiththeusageofharmonicfunctions is suggested. In the mentioned publica-tions,itisshownthatthetransitionsareasofcontig-uous zones arewell described by generalized equi-libriumequation (2).Theseequationsareknown in

literature[8]…[11]andallowstoobtainrealfieldsoftangentialstress in theareas,wherestresseschangetheiroperator.

Aftertransformationsfromtheequilibriumequa-tiononemayobtainthesystemconnectingtangentialstresswiththeremainderofnormalones[4],wewillhave

,

,(6)

.Expression (6) may be used if there exist the

possibilitytoconnectnormalandresidualstresses.Tosimplify(6)letususethelimitations,whichare

determinedbykinematicproblemspecification. Inthiscase,mixedderivativesfromtangentialstressesaccordingtothecoordinatesareequaltozero

hence

Tosatisfyboundaryconditionsinthestressesandacceptanceofpossiblesimplifications,wewillhave

(7)

Thesystem

, ,

.(8)

Thesystem(8)isrepresentedasthedeterminingoneinthesystemofequationsofplasticitytheory.

Connection of normal tangential stresses may bedefinedusingequationsofconnection(3)

.(9)

Duringproblemstatement(1)...(5),wewillhave

, ,


Materials science

.(10)

Such schemes simplify the solution, as in theequation thereusedonedecisionfunction τ .Con-

sideringin(10)theproductswithgoniometricfunc-tions[4]...[7],onemayaccept

,

,(11)herein

111 FctgF Α= , 222 FctgF Α= , 333 FctgF Α= .

Howeverrealconnectionofnormalandtangentialstressesforspaceproblemdonotcorrespondto thecorrelations(9)...(11).Thisconnectionismorecom-plicated,whichmaybeseenduringelementaryanal-ysisofsolutionofequilibriumequation.Thereraisesthe question,when it is possible to use differentialequations(2)or(10)inthesolutionofspaceproblem.

SolutionofthetaskDeterminationofcomponentsofstresstensorThepeculiarityofthissolutionistheuseofcom-

bination of flat functions for components of stress

tensor,which is determined by the requiring of thesolutionofclosedtaskofplasticitytheory.Thisisex-plainedundertheconditionwhenthesolutionsatis-fiesallthesystemofequationsofplasticitytheoryintheformof(1)...(5),qualitativelyandquantitativelycorrectcharacterizestransitionareasofplasticzonesinvariousprocessesofmetal treatmentunderpres-sure.

According toworks [4]...[7],[12]and(7)...(11),there used trigonometric and fundamental substitu-tionsofthefollowingform

,

,(12)

Where −θ'³ -aretheargumentsofgoniometric

functionsdependingonthecoordinatesX, Y, Z, de-finedbythetasksolution;

−θ'³ are theexponentsdependingon thecoor-

dinatesX, Y, Z, definedbythetasksolution.Duringsubstitution(12)into(10),wewillobtainthesystemofequations

Thefirstequation

Thesecondequation

(13)

Thethirdequation


Materials science

Intheequations(13)thereappearoperatorsbeforegoniometricfunctions.Ineachoperatorthereappear

similarbrackets.Thesolutionrunsforwardwhenthebrackets

)( 11'1 yx FΑ+θ , )( 11

'1 xy FΑ−θ , ( )zy F22

'2 Α+θ ,

)Α+( xz F333θ , )Α+( xz F333θ , yx F11'1 Α−=θ

areequaltozero,theequationsturnintoidenticalequations.Letusshowthisontheexampleofthefirstequationofthesystem(13)

yx F11'1 Α−=θ , xy F11

'1 Α=θ . (14)

Theseconddifferentialcoefficients

. (15)

From the correlations (14) and (15)we come tothe Laplace equations, which are satisfied by thefunctions '

1θ and 11FΑ

.(16)

Itisdefinedtheclassofaddedintoreviewfunc-tions–theyarebalanced.Undersuchstatementtheyareconsideredtobeknown.Fromtheotherpointofview,correlations(14)and(15) turn(13) into iden-tical equations. In such a way, functions τ may besolvedasfollows

(17)when

Normal stresses yx σσ ,, zσ are determined ac-

cordingtotheknowntangentialstresses(17)from

theequilibriumequation.Takingintoaccountdevi-atoriccomponentwewillhave

CzyfFCoskFCoskx +++Α⋅−Α⋅= ),()()( 033'3111

' σσ ,

CzxfFCoskFCosky +++Α⋅+Α⋅−= ),()()( 0222111

' σσ ,(18)


Materials science

)()()(2 111333222 FCoskFCoskFCoskzy Α⋅−Α⋅−Α⋅⋅=′−′ σσ .Using(18)letusreturntothedifferenceofnormalstressesinequations(8)

)()()(2 111333222 FCoskFCoskFCoskzy Α⋅−Α⋅−Α⋅⋅=′−′ σσ ,

)()()(2 111333222 FCoskFCoskFCoskzy Α⋅−Α⋅−Α⋅⋅=′−′ σσ ,(19)

)()()(2 222111333 FCoskFCoskFCoskxz Α⋅−Α⋅−Α⋅⋅=′−′ σσ .

Thegivendifferencesrepresentthecombinationof flat functions,more complicated than (9). It ispossible to show that mixed derivatives from the

differences in (9)and in (19)give thesameresultandthatiswhyitacceptabletouseflatfunctionsforstresses.Letusshowit:

Planeproblem( ) −Α⋅Α⋅Α⋅−=Α

∂∂∂

⋅=⋅∂∂∂

111111111111 222 FCosFFkkFCoskyx

Fyx yxxyxyτ

( ) 11111111112 FSinFkFkFk xyyxxy Α⋅Α+Α+Α−,

Spaceproblem

( ) ( +Α+Α−Α⋅Α⋅Α⋅−= yxxyyxxy FkFkFCosFFkk 1111111111111 22

( ) ( +Α+Α−Α⋅Α⋅Α⋅−= yxxyyxxy FkFkFCosFFkk 1111111111111 22

) 11111 FSinFk xy ΑΑ+

Shown peculiarity is the principal detail in thepresentedsolution.Takingintoaccounttheoperator

oftheexponent,thefollowingsolutionsarepossible.Stresstensorcomponentsofspaceproblem

(20)

when


Materials science

Herein

.Qualitativesolutionfordeformationrateingenerallookslike

,

(21)

.Theconditionofvolumeconstancyissatisfied

0=++ zyx ξξξ.

In the equations of connection the arguments ofgoniometric functions have the same functions F.Hereinthefunctionswithindex1aredeterminedbythecoordinatesXY,2 -YZ,3 -ZX. In the formulas(21) ,1β 3β 3β aretheunknownvariables.Letus

substitute deformation rates (21) into thefirst threedifferentialcontinuityequationsofdeformationrates(4a), than after reduction and simplifications theequationscometo

xyFSin

xFCos

yFCos

∂∂Α⋅∂

⋅=∂

Α∂−

∂Α⋅∂ )(2)()( 111

2

2111

2

2111

2 βββ

,

yzFSin

yFCos

zFCos

∂∂Α⋅∂

⋅=∂

Α⋅∂−

∂Α⋅∂ )(2)()( 222

2

2222

2

2222

2 βββ

,(22)

zxFSin

zFCos

xFCos

∂∂Α⋅∂

⋅=∂

Α⋅∂−

∂Α⋅∂ )(

2)()( 333

2

2333

2

2333

2 βββ

.Despitethecomplexcombinationoffunctionsin

theequationsofthesystem(22)therearesimilarun-knownvariablesβ ,enterthesystemofequationsin

the1stdegree.Thisallowstousefundamentalsubsti-tutionasfollows[12]:

Determination of components of rate of defor-mation tensor

Kinematicpartofthetaskisapeculiarkindoftestforthestationaryone,asthroughtheequationofcon-nectionthereformedrestrictionsonstressfunctions.Tosatisfythesecondpartofthesystemofdifferentialequations(4b),thesimplestvariantistheone,where

mixedderivativesfromshearspeedofdeformationsintheleftpartsequalzero.Acceptableare:

linear speeds of deformations and normal stressesaswell,representthecombinationofflatfunctions.Thefirstequationofconnection(3)willbeasfol-lows

.Fromthelastexpressionitfollowsthatthespeed

ofdeformations in the rightpart,undercertaincor-relationsoftheinputfunctions,isequalto

)()( 333111 FCosFCosx Β⋅−Β⋅= ββξ , )()( 222111 FCosFCosy Β⋅+Β⋅−= ββξ ,.


Materials science

.(23)

Substituting(23)into(22)wewillobtainthesystemThefirstequation

,Thesecondequation

,Thethirdequation

As in the case of (13) the system (24) may besolvedifthereisapossibilitytoavoidthenonlinear

effect.Letus taking thebrackets inoperatorsequalzero,than

yx F11''1 Β−=θ , xy F11

''1 Β=θ ; yz F22

''2 Β=θ

, yz F22''2 Β=θ ;

xz F33''

3 Β−=θ , yx F11''1 Β−=θ .(25)

Theseconddifferentialcoefficients

Input functions''

iθ and ii FΒ are theharmonicones,andthefunctionsforstresstensorcomponentsaswell.

The expression for determination tensor compo-nentsofdeformationspeedforspacetask,takingintoaccount(21)…(25),looksasfollows

(26)

(24)


Materials science

CzyfFCoskFCoskx +++Α⋅−Α⋅= ),()()( 033'3111

' σσ ,CzxfFCoskFCosky +++Α⋅+Α⋅−= ),()()( 0222111

' σσ,

CyxfFCoskFCoskz +++Α⋅−Α⋅= ),()()( 0222333' σσ .

Differencesofnormalstresses)()()(2 333222111 FCoskFCoskFCoskyx Α⋅−Α⋅−Α⋅⋅=′−′ σσ

,)()()(2 111333222 FCoskFCoskFCoskzy Α⋅−Α⋅−Α⋅⋅=′−′ σσ

,−Α⋅⋅Α−Α⋅⋅Α−++⋅=′ 222111333111

23

22

213 FCoskFCoskFCoskFCoskkkkiσ .

Substitutinginto(1)wewillobtain−Α⋅⋅Α−Α⋅⋅Α−++⋅=′ 222111333111

23

22

213 FCoskFCoskFCoskFCoskkkkiσ

222333 FCoskFCosk Α⋅⋅Α−. (27)

Itshouldbementionedthatundertheradicalthereis the sumof squaresofflat functions forvalues k.Appearanceofdifferenceofgoniometricfunctionsmakessignificantallowancestotheresultandaffects

the integral characteristic of trigonometric compo-nent.At 02 =k theexpression(27)issimplifiedandlooksasfollows

33311123

21i CoskCoskkk3 ΦΑ⋅⋅ΦΑ−+⋅=σ′

.

If 032 == kk ,wehaveaflattaskofplasticitytheory,than

T121i k3k3 σ=⋅=⋅=σ′ ,т.е. 31

Tk σ=

.Variantwhenallthestressesmaybeshiftedtothenegativezoneontheaccountofmeanstress 0σ looksas

follows

,

when

Integral characteristic of stress state From the above mentioned it follows that the

closedsetandsolutionof the taskofplasticity the-orydoesnotcomplicateit,andatcertainapproaches–simplifiesandallowstofindthesolutionofspace

probleminanalytical form. Tofindgeneralized in-dexesofstressstateofthepointwiththeuseofsolu-tion (22) is of some interest. Let us consider somevariantsrepresentedbycorrelations(22).Letus


Materials science

The sum of squares of values k remains un-changed.Whencompareexpressions (27)and (28),one may see that the operators before goniometricfunctionschange.Thisleadtothechangesincalcula-tions.Incaseof(27),intensitymaybegreaterthanincaseof(28).Stabilityofsumofsquareskatvariouscombinationsofstressestestifiesthattheintensityof

stresseschangesnearthisvaluetothebiggerorsmall-ersiteandvalue

k3kkk3 23

22

21i ⋅=++⋅=σ ′′ ,

isapeculiarkindofcoreofsolution,At 01 =k theexpression(28)issimplifiedandturnsto

.If 021 == kk ,wewillobtaintheflattaskofplasticitytheory,than

T323i k3k3 σ=⋅=⋅=σ ′′ ,т.е. 33

Tk σ= .

From the analysis of integral characteristics ofstress state, it is seen that the result of space task,during simplifying, may be compared with knowndependences for flat task. This allows to considerobtainedsolutionatvolume loadingasmoregener-alandmorecomplexcharacteristicofstressstateofthepoint,satisfyingtheclosedsystemofequationsofplasticitytheory.

Conclusions 1. Integral characteristics of stress state deter-

minemechanicalcharacteristicsofplasticme-dium.Presentedcorrelationmaybeused foraccountofnonuniformityofmechanicalprop-ertiesofprocessedmaterial.

2. Theresetclosedvolumetaskofplasticitythe-ory (static and kinematic part) allowing onsomeextentnottocomplicatethesolutionbuttosimplifyit.

3. Correction factor of simple solution is kine-maticpartofthetask,whichiseasiertosatisfy

bythecombinationsofflatfunctions.4. Theobtainedresultisinsomecorrespondence

withsolutionofflattaskandmayberecom-mended for determination of components ofstresstensor,deformationtensorandgeneral-izedindexesofpointstate.

References1. Chigirinskiy V.V., Sheyko S.P., Echin S.M.

Prostranstvennayazadachateoriiplastichnosti[Space problem of plasticity theory]. Collec-tionofscientificpapers,Metaltreatmentunderpressure. Kramatorsk. 2013, No 2 (35), p.p.3-8.

2.KapturovL.E.Kontaktnyiesilyivochagedefor-matsii pri prokatke polos [Contact forces in roll passes of strips].Materials ofAll-Unionscientificandtechnicalconference“Theoreti-calproblemsofrollingindustry”.Dnepropet-rovsk,1975,p.p.428-432.

Differenceofnormalstresses

Substitutingintothestressintensity,wewillobtain+Α⋅⋅Α+Α⋅⋅Α−++⋅=′′ 222111333111

23

22

213 FCoskFCoskFCoskFCoskkkkiσ

222333 FCoskFCosk Α⋅⋅Α+.

(28)


3. Chigirinskiy V.V. O novyih podhodah resh-eniya zadach teorii plastichnosti [About new approaches of task solution in the plasticity theory].Collectionofscientificpapers,Metaltreatment under pressure. Kramatorsk. 2009,No1(20),p.p.41-49.

4.ChigirinskiyV.V.,MazurV.L.,BergemanG.V.,Legotkin G.I., Slepyinin A.G., ShevchenkoT.G. Proizvodstvovyisokoeffektivnogomet-alloprokata [Manufacturing of high efficiency of rolled stock].Dnepropetrovsk,DnIpro-VAL,2006,p.265.

5.ChigirinskiyV.V.Novoereshenieploskoyzad-achi teorii plastichnosti [New solution of flat task of plasticity theory]. Scientificworks ofDonNTU,Metallurgiya,No10(141).Donetsk.2008,p.p.105-115.

6.ChigirinskiyV.V. (2009).Metod resheniya za-dachteoriiplastichnostis ispolzovaniemgar-monicheskih funktsiy [Problem solving tech-niqueofplasticitytheorywiththeuseofhar-monicfunctions].Chernaya metallurgiya.No5,p.p.11-16.

7.ChigirinskiyV.V.Nekotoryieosobennostiteorii

plastichnostiprimenitelnokprotsessamOMD[Somepeculiaritiesofplasticitytheoryinthecontext of metal treatment under pressure].Works of scientific and technical conference «Teoriya I tehnologiya protsessov plastich-eskoy deformatsii-96».Moscow,MISiS,1997,p.p.568-572.

8.PrantlL.Z.A.M.M.1923,p.401.9.NadaiA.PlastichnostIrazrushenietverdyihtel

[Plasticity and destruction of solid bodies]. Moscow, Publishing house of foreign litera-ture,1954,647p.

10. BezuhovN.I.Osnovyi teorii uprugosti, plas-tichnostiipolzuchesti[Fundamentals of elas-ticity, plasticity and creeping theory]. Vyis-shayashkola,1968,498p.

11.ArkulisG.E.,DorogobidV.G.Teoriyaplastich-nosti[Plasticitytheory].Moscow,Metallurgi-ya,1987,351p.

12. Tihonov A.N., Samarskiy A.A. Uravneniyamatematicheskoy fiziki [Equations of mathe-matical physics].Moscow,Nauka,1977,735p.

Materials science

Analysis and modelling of complex rheologic mediums in conditions of thermomechanical loading

UDC539.374.6

Chigirinsky V.V.

D.Sc. in engineering, professor,Head of ZNTU Metal Forming Department

Zaporizhzhya National Technical University,Zaporozhye, Ukraine



Documents

UDC 539.374.1 Determination of integral …...Determination of integral characteristics of stress state of the point during plastic deformation in conditions of volume loading UDC