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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: valerij@zntu.edu.ua
Lenok A. A.
Assistant of ZNTU Metal Forming Department
Zaporozhye National Technical University,Zaporozhye, Ukraine
E-mail: anastasion4@rambler.ru
Echin S. M.
Assistant of ZNTU Metal Forming Department
Zaporozhye National Technical University,Zaporozhye, Ukraine
E-mail: ser.echin92@gmail.com
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
155Metallurgical and Mining IndustryNo.11 — 2015
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
Metallurgical and Mining Industry156 No.11 — 2015
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
, ,
157Metallurgical and Mining IndustryNo.11 — 2015
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
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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)
159Metallurgical and Mining IndustryNo.11 — 2015
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
Metallurgical and Mining Industry160 No.11 — 2015
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 ,.
161Metallurgical and Mining IndustryNo.11 — 2015
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)
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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
163Metallurgical and Mining IndustryNo.11 — 2015
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)
Metallurgical and Mining Industry164 No.11 — 2015
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.
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[Plasticity and destruction of solid bodies]. Moscow, Publishing house of foreign litera-ture,1954,647p.
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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
E-mail: valerij@zntu.edu.ua
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