Osnovni koncept numerične simulacije radialnega
kovanja
Basic Concepts of Numerical Simulation of a Radial
Forging Process
T. Rodič*, D. R. J. Owen**	UDK: 621.73.045:519.6
ASM/SLA: F22, Q24,1-66, U4g, U4k
1. UVOD
S spoznanji splošnih principov fizikalne metalurgije1 postajajo preoblikovalni procesi v vročem pomembnejši. Preoblikovanje v vročem že dolgo ni več samo spreminjanje oblike preoblikovanca, temveč termomehanska obdelava materiala, ki naj privede do ugodnih strukturnih sprememb. Pri upoštevanju medsebojnih odvisnosti med strukturo, lastnostmi in obnašanjem materiala med plastičnim preoblikovanjem so deformacija, hitrost deformacije in temperatura tiste fizikalne večine, ki imajo odločilni vpliv. Nadzorovana porazdelitev teh termo-mehanskih parametrov med preoblikovanjem je potrebna pri optimiranju preoblikovalne operacije. Preoblikovalni procesi v vročem so zahtevni za eksperimentalna opazovanja zaradi visokih temperatur in preoblikovalnih hitrosti. Od tod tudi potreba po matematičnih in numeričnih modelih, ki pripomorejo k boljšemu razumevanju eksperimentalnih rezultatov ali celo delno nadomeščajo draga preizkušanja. Tri klasične metode za analizo preoblikovalnih procesov so bile pogosto uporabljane v preteklosti2:
—	metoda elementarne plastomehanike
—	metoda drsnih linij
—	metoda zgornje in spodnje meje.
Analiza preoblikovalnih procesov je zahtevna in mnogo poenostavitvenih predpostavk je bilo vpeljanih v klasičnih metodah, da bi se izognili matematičnim težavam, kar je seveda zmanjševalo njihovo uporabnost. Napredek numeričnih metod v zadnjem času, posebej metode končnih elementov3 (MKE) in vzporedno zmanjševanje cen računalniških obdelav, ponuja možnost za realnejše simulacije preoblikovalnih procesov.
2. MATEMATIČNI MODEL
Za analizo porazdelitev napetosti, deformacij in
temperature, ki se spreminjajo znotraj deformacijske cone med preoblikovanjem, je nujna uporaba numeričnih metod. Razvoj MKE na področju plastomehanike4 in prenosa toplote ponuja zadovoljivo orodje za računalniško simulacijo preoblikovalnih procesov v vročem.
Simulacijo preoblikovalnega procesa v vročem lahko idealiziramo5, kot je to prikazano na sliki 1.
1. INTRODUCTION
Since the general principles of physical metallurgy were recognised1, the hot vvorking processes are no longer only concerned with shape changes but also con-sider the thermomechanical treatment vvhich contributes to beneficial structural changes vvithin the material. In considering the interaction betvveen the structure, properties and performance of the material under plastic deformation, the strain, strain rate and temperature are quantities vvhich have a fundamental influence and a controlled variation of these thermomechanical parame-ters is essential for optimising the forming operation. The nature of hot vvorking processes makes experimen-tal observations difficult, due to the high temperatures and speeds involved. Therefore mathematical and numerical models have a role to play in either improving the interpretation of experimental results or even replac-ing, in part, an expensive testing programme. Three classical methods for analysing metal forming problems have been widely used in the past2:
—	elementary plasticity
—	the slip line method
—	the upper and lovver bound method.
Metal forming processes are complex and many sim-plifying assumptions have been introduced to these classical methods in order to avoid mathematical diffi-culties. This, hovvever, limits their applicability. Recent developments of numerical methods, in particular the Fi-nite Element Method3 (FEM), and a parallel reduction in unit computing costs offer an opportunity for a more realistic simulation of vvorking processes.
2. MATHEMATICAL MODEL
The complex stress-strain and temperature distribu-tions vvhich vary across the deformed region during the deformation process require the use of numerical methods. Developments in the FEM in the field of plastome-chanics4 and heat transfer offer a satisfactory tool for computer simulation of hot vvorking processes.
The numerical simulation of the hot vvorking process can be idealised5 as shovvn in Fig. 1.
* FNT — Odsek za metalurgijo, Univerza E. Kardelja, Ljubljana
** Dept. of Civil Engineering, University of VVales, Swansea, U. K.
Slika 1:
Simulacija preoblikovalnega procesa.
Vhodni podatki predstavljajo lastnosti in začetno stanje preoblikovanca, kot so: oblika, porazdelitev temperature, sestava in mikrostruktura materiala.
Matematična simulacija preoblikovalnega procesa: Pri MKE razdelimo preoblikovanec na manjša območja, imenovana elementi. Togost vsakega elementa je določena z njegovo geometrijo in lastnostmi materiala, ki jih elementu pripišemo. Oba vpliva obravnavamo ločeno in zato relativno enostavno vgrajujemo različne materialne modele. Model preoblikovanca dobimo s sestavljanjem togostnih matrik elementov. Takšen pristop omogoča analizo različnih geometrijsko zahtevnih preoblikovalnih procesov.
Z robnimi pogoji simuliramo različne pogoje, v katerih poteka proces.
Izhodni rezultati. Po obdelavi rezultatov numerične analize določimo optimalne tehnološke pogoje. Nazoren grafični prikaz rezultatov je pomemben sestavni del analize z MKE.
3. MATERIALNI MODEL
Pri modeliranju obnašanja materiala med plastično deformacijo je potrebno poznavanje ustrezne napetosti tečenja. V splošnem je ta odvisna od sestave in mikro-strukture materiala in od hitrosti deformacije, temperature ter deformacijskega stanja, povzročenega s plastično deformacijo. Napetost tečenja določimo z nateznim, tlačnim ali torzijskim preizkusom6. Pogosto napetosti tečenja niso dosegljive za specifične kombinacije termome-
Slika 2:
Ploskve napetosti tečenja v prostoru termomehanskih parametrov. Fig. 2:
Flow stress surfaces in the space of the thermomechanical par-ameters.
Fig. 1:
Simulation of the vvorking process.
Input data represent the material properties and in-itial state of the vvorkpiece; such as shape, temperature distribution, composition and microstructure of the material.
Mathematical simulation of the vvorking process: In FEM the workpiece is divided into small regions termed elements. The stiffness of each element is determined by its geometry and material properties. Both effects are considered separately and therefore it is relatively sim-ple to incorporate different material models. The vvorkpiece model is obtained by combining the stiffness con-tribution of each element. Thus any complex shape of the vvorkpiece model can be analysed using the FEM.
Boundary conditions are applied to simulate different conditions under vvhich the process is to operate.
Output results: A decision on the most suitable set of operating conditions can be made by postprocessing the numerical results. Graphical representations play an integral part in the interpretation of the numerical results of a FEM analysis.
3. MATERIAL MODELS
In the modelling of material behaviour during metal-vvorking processes a knovvledge of the appropriate flovv stress for the material is essental. In general this will de-pend on the composition and microstructure of the material and on the strain rate, temperature and deforma-tion modes imposed by the vvorking process. The flovv
		0	
	0d=0-0p H		\op O. > U
		(u > w	
77777777777777
Slika 3:
Osnovni enodimenzionalni elasto-viskoplastični reološki model.
Fig.3:
Basic one-dimensional elasto-viscoplastic rheological model.
hanskih parametrov in jih ocenimo s pomočjo poznanih podatkov. Znane so različne interpolacijske enačbe, kot na primer Hajdukova7 ali Sellars-Tegartova enačba8. Na podlagi teh enačb lahko določimo potencial napetosti tečenja v prostoru deformacije, hitrosti deformacije in temperature5 (SI. 2). Ploskev A-B-C-D je določena z (p, <p, T, ki povzročajo enako napetost tečenja Kf^r'\ določenem stanju mikrostrukture materiala. V splošnem se v delcu materiala med preoblikovanim procesom termo-mehanski parametri spreminjajo iz (p, (p, T\ <f/, ep/, T' in temu ustrezno se spremeni napetost tečenja iz K, v K'h ki je določena s ploskvijo A'-B'-C'-D'. Za napovedovanje sprememb termomehanskih parametrov med preoblikovalnim procesom v vročem uporabljamo elasto-viskoplastični materialni model, združen s prenosom toplote.
4.	OSNOVNE ENAČBE ELASTO-VISKOPLASTIČNOSTI
Osnovni enodimenzionalni elasto-viskoplastični reološki model, ki je predstavljen na sliki 3, lahko razširimo za primer splošnega kontinuuma. Postopek je podrobno opisan v literaturi [4]. Na tem mestu bodo predstavljene le najosnovnejše enačbe (SI. 4).
Hitrost deformacije je razdeljena na elastično ef in visokoplastično £v.„ komponento (En. 1). Elastična hitrost deformacije je določena s Hookeovim zakonom, medtem ko erp izrazimo z ustreznim zakonom tečenja. S tem dobimo enačbo (En. 2), kjer je y parameter tečenja in 0 je funkcija, ki je različna od nič samo za pozitivne vrednosti funkcije F. Začetek visokoplastičnega obnašanja določa pogoj, izražen s skalarno enačbo (En. 3), kjer je A, napetost tečenja pri enoosnem preizkusu in x parameter utrjevanja. Enačbi (En. 1) in (En. 2) lahko zapišemo v inkrementalni obliki in tako dobimo izraz (En. 4), ki določa spremembo napetosti v časovnem koraku A t„ = tn+,— t„. Z uporabo implicitne časovno integracijske sheme (En. 6), kjer je hitrost viskopolastične deformacije na koncu časovnega intervala izražena s pomočjo prvih dveh članov Taylorjeve vrste (En. 7), dobimo končni izraz za inkrement napetosti. (En. 8,9 ). Plastično delo se med preoblikovanjem spreminja v toploto, kar povzroča prirastek temperature. Povprečno hitrost generacije toplote v časovnem koraku izračunamo s pomočjo izrazov (En. 10, 11), kjer je f frakcija plastičnega dela, ki jo akumulira material.
Analiza nestacionarnega prenosa toplote z MKE je opisana v literaturi [9].
5.	ROBNI POGOJI
Z robnimi pogoji simuliramo pogoje, v katerih poteka preoblikovalni proces. V splošnem ločimo mehanske in termalne robne pogoje.
S.l. Mehanski robni pogoji
Mehanski robni pogoji so odvisni od oblike in gibanja orodja ter od trenjskih razmer med orodjem in pre-oblikovancem.
Med preoblikovanjem se preoblikovalni stroj elastično deformira. Problem poenostavimo, če privzamemo, da je stroj tog. Tako je gibanje orodja popolnoma popisano z gibanjem ene točke, ki ga izračunamo iz kine-matike stroja10. Ti podatki so del vhodnih podatkov in so lahko časovno odvisni.
stress is determined from true stress — true strain data obtained from tension, compression or torsion tests6. Frequently stress — strain data are not availabie for specific combinations of the required conditions and they must be estimated from data that are availabie. Various expressions have been suggested for the interpola-tion of high temperature data, for example Hajduk's7 re-lation or the Sellars — Tegart expression8. On the basis of these expressions the material flow stress surface in the strain, strain rate and temperature space can be re-presented5 (Fig. 2). Surface A-B-C-D is defined by the (p, <p, 7required to give the same flow stress K, for a parti-cular state of the microstructure of the material. In a part of the deformed material the thermomechanical parame-ters (p, (p, fvvill change to cpf, (//, T' during the forming process and the flow stress will be changed from K, to KI defined by surface A'-B'-C'-D'. To prediet the change of the thermomechanical parameters during the hot vvorking operation an elasto-viscoplastic material model coupled vvith heat transfer is used.
4	BASIC CONCEPTS OF ELASTO-VISCOPLASTICITY
The basic one dimensional elasto-viscoplastic rhe-ological model shovvn in Fig. 3 can be extended to the čase of general continua. Details of this approach are provided in Ref. 4 and only essential expressions are reproduced in Fig. 4.
The total strain rate is separated into elastic ee and viscoplastic Eyp components (Eq. 1). The elastic strain rate obeys Hooke's law and evp is expressed by the ap-propriate viscoplastic flow rule. This gives the governing equation (Eq. 2) vvhere y is fluidity parameter and O is taken as non-zero for positive values of the yield func-tion, F, only. The onset of viscoplastic behaviour is go-verned by a scalar yield condition (Eq. 3) vvhere the un-iaxial yield stress is denoted by K, and j is a hardening parameter. The rate equations (Eq. 1, 2) can be vvritten in an ineremental form to give the stress inerement oc-curing in tirne step A tn= tn+tn (Eq. 4). Use of the im-plicit tirne integration sheme (Eq. 6) vvhere the viscoplastic strain rate at the end of the tirne step is predieted by a limited Taylor series expansion (Eq. 7) resultes in (Eq. 8, 9) for the stress inerement. Most of the plastic vvork done during the tirne step is converted into heat and gives an inerease of temperature. The average rate of heat generation vvithin the time step vvhich enters the thermal analysis is calculated according to (Eq. 10, 11) vvhere, f, is a fraetion of the plastic vvork stored in material.
Thermal transient finite element analysis is deseribed in Ref. 9.
5	BOUNDARY CONDITIONS
The boundary conditions represent the conditions under vvhich the process is to operate. Generally we dis-tinguish betvveen mechanical and thermal boundary conditions.
5.1. Mechanical boundary conditions
The mechanical boundary conditions depend on the shape and movement of the die and the frietion at the die — vvorkpiece interface.
During the course of the forming process the forming machine undergoes elastic deformation. To simplify the problem the forming machine is assumed to be rigid. Therefore the motion of the die can be completely deseribed by specifying a velocity at one point vvhich can be calculated from the kinematics of the forging ma-
OSNOVNE ENAČBE ELASTO-VISCOPLASTIČNOSTI BASIC EQUATIONS OF ELASTO-VISCOPLASTICITY
{ * } - { *e } + { <vP) ...............................(D
{ i } - [ E J"1 {*}+ 7 «t(F)>
3F
3{<t)
(2)
F((ff),x) - f({<Tl) - Kf(x) " 0
(3)
Inkrementalna oblika osnovnih enačb: Incremental form of the basic equations:
M" [ * ][**}-[ E ] { *<»} - { ^vp}
. .(4)
Ae"} - [ B" ] { AU" } ...............................(5)
] - At„ [ (1-0) + 9 ^vp+1]
; 0 4 9 4 1 ____(6)
. n+ll
fvp j
{ žvp }
+ 9'^vp) A<jn
3{<xn)
(7)
A."} - [ D" ] [ Bn ] { AUn| - { <v$}
(8)
[ °n]- [ E I"'" 64tn [ ^rH1 1
3(
3{ff )
(9)
Povprečna hitrost generacije toplote v časovnem koraku Average rate of heat generation within the tirne step:
Qn- (1-f) { ,n+a}T 1 { en,l} . { 4y„ }
(10)
{ an+a} = { a n } + a { Aan } ; O L a 6 1
(H)
Slika 4:
Pregled osnovnih enačb. Fig. 4:
Owerview of basic equations.
Proces radialnega kovanja v vročem poteka brez maziv pri visokih temperaturah, zato so trenjske razmere v stiku med orodjem in preoblikovancem zahtevne za numerično obravnavo. Porazdelitev strižnih napetosti v kontaktu lahko ustreza Coulombovemu zakonu, pravilu o konstantni strižni napetosti ali lepljenju. Smer strižnih napetosti je odvisna od relativnega gibanja med orodjem in preoblikovancem. Ker je relativno gibanje materiala odvisno od trenjskih razmer, je problem očitno nelinearen. Mesto nevtralne točke, kjer ni relativnega gibanja, najdemo z iteracijskim procesom. Ko najdemo pozicijo nevtralne točke, lahko trenjske razmere predpišemo, kot je to prikazano na sliki 5.
chine.'0 These velocity data are part of the input and can be varied vvith tirne.
Since the radial forging operation is carried out at high temperatures vvithout lubrication the friction condi-tions at the tool-vvorkpiece interface face are complex. The distribution of the shear stress at the interface may obey either Columb's lavv, the constant shear rule or sticking friction conditions. However, the direction of the shear stress is unknovvn due to relative movement of the material at the interface. Since the relative movement is friction dependent the problem clearly becomes nonli-near. The position of the neutral point, vvhere relative sliding is zero must be found by an iterative procedure. After the position of neutral point is established the friction conditions can be prescribed as presented in Fig. 5.

vvorkpiece preobli kovanec
Tool orodje
, neutral point nevtralna točka

--~
Slika 5:
Predpostavljene trenjske razmere. Fig. 5:
Assumed friction conditions.
5.2. Termalni robni pogoji
Med preoblikovanjem preoblikovanec izgublja toploto zaradi stika s hladnejšim orodjem in s sevanjem. Za opisovanje teh pogojev uporabljamo standardne matematične robne pogoje.
—	predpisano temperaturo na stični površini
—	predpisan toplotni tok skozi stično površino
—	Newtonov zakon o prenosu toplote
6. ILUSTRATIVNI PRIMER
Z MKE smo analizirali proces radialnega kovanja na kovaškem stroju z zaokroženimi kladivi, ki je shematično prikazano na sliki 6. Napetostno-deformacijsko stanje pri takšnem kovanju je kvaziaksisimetrično. Geometrijo delovnih površin kovaškega kladiva razdelimo na tri dele:"-12
I — vhodno cono II — kalibrimo cono in III — izhodno cono.
Skoraj vsa plastična deformacija nastopi v vhodni coni. Glavna parametra vhodne cone sta kot a na kladivu ter povprečna vhodna hitrost Vinp materiala. Za kovanje z redukcijo 0 /20/0 80 mm, ki smo ga analizirali, je bil kot a= 10°in Vinp= 45 mm/s. Iz teh podatkov in iz poznane kinematike stroja lahko izračunamo:
Ld projekcijo vhodne dotikalne površine na os
simetrije (113.4 mm) Uinp pomik vhodnega materiala v smeri simetrijske
osi po vsakem udarcu (10 mm) Wrad delovni hod kladiva v radialni smeri
(1.76 mm) tud čas trajanja udarca (0.018 s)
Uporabljeni podatki o materialu13:
Material: X5CrNil8.8 Začetna temperatura 1000° C
5.2. Thermal boundary conditions
During the forming process the vvorkpiece loses heat due to contact vvith the colder die and also by radiation. To model these effects, standard mathematical boun-dary conditions are used:
—	prescribed temperature at the contact surface
—	prescribed flux across the contact surface
—	Newton's lavv of heat transfer
6 ILLUSTRATIVE EXAMPLE
A roundfaced die radial forming proces vvhich is schematically presented in Fig. 6, has been analysed us-ing FEM. The stress-strain field is assumed to be quasi-axisymmetric in this čase. The geometry of the vvorking surfaces of the radial forging die is divided into three parts:1112
I	— the inlet cone
II	— the sizing cone
III	— the outlet cone
Nearly ali the plastic deformation occurs in the inlet cone. The main parameters of the inlet cone are the angle a of the die and the average speed Vinp of the input material. In the present example the reduction vvas 0 120/0 80 mm and the inlet cone angle a and speed Vinp vvere 10° and 45 mm/s respectively. From this data and kinematics of the machine the follovving boundary conditions are evaluated:
Ld projection of the inlet cone contact area on to
the centreline axis (113.4 mm) Uinp displacement of the input material along the
centraline axis after each punch (10 mm) Wrad displacement of the die in the radial direction
vvhile in contact vvith the vvorkpiece (1.76 mm) tud duration of the punch (0.018 s)
The material properties used in this example are giv-en belovv:
Material: X5CrNi18.9 Initial temperature 1000°C
Slika 6:
Shematični prikaz procesa radialnega kovanja. Fig. 6:
Schematical view of the radial forging process.
nja
Lastnosti materiala pri začetni temperaturi kova-
.13
Material properties at the initial temperature:
modul elastičnosti Poissonov količnik gostota
specifična toplotna kapaciteta
120 kN/mm2 0.34 7430 kg/m3 650 J/kg K
—	Young's modulus
—	Poisson's ratio
—	density
—	specific heat capacity
120 kN/mm2 0.34 7430 kg/m3 650 J/kg K
Interpolacijska funkcija za napetost tečenja: K, = Kfo • A i ■ e ~ m'T • A2 • (p"" • Ar (pmj
kjer je
Kf0= 189.5 N/mm2 A, = 12.997 m, =0.00258 A2= 1.570 m2 = 0.196 A3 = 0.740 m3 = 0.128
Uporabljeni so bili 8-vozliščni Serendipity aksialno simetrični končni elementi z reducirano (2 x 2) numeri-čno integracijsko shemo14 ter von Misesovim kriterijem tečenja z izotropnim modelom utrjevanja materiala/ Nelinearni elasto-viskoplastični problem smo reševali z metodo tangencialne togosti. Pri tem smo uporabili implicitno časovno integracijsko shemo s korekcijo napetosti na koncu vsakega časovnega koraka15. Sistem linearnih enačb smo reševali s frontalno metodo16.
Interpolation function for high temperature data:7 K, = Kto • A, • e-'miT • A2 • (pmz • A3 • cpm3
vvith:
Kfo= 189.5 N/mm2
A, = 12.997 A2= 1.570 A3 = 0.740
m, =0.00258 m2 = 0.196 m3 = 0.128
The 8-noded Serendipity axisymmetric elements vvith reduced (2 x 2) numerical integration scheme14 vvere used. A von Mises yield function vvith isotropic strain hardening was employed.4 The nonlinear viscoplastic problem was solved by tangential stiffness solution algo-rithm. An implicit tirne integration scheme vvith the stress correction at the end of each tirne step vvas per-formed.15 The resulting set of linear equations vvere solved by the frontal technique.16
REZULTATI
Na slikah 1,8 in 9 so prikazane posamezne faze razvoja prirastka plastičnih deformacij v preoblikovalni coni med udarcem kladiva. S pomočjo rezultatov MKE lahko na enak način predstavimo prostorsko in časovno porazdelitev ostalih termomehanskih parametrov med preoblikovanjem. To so temperatura ter vse komponente in primerjalne vrednosti tenzorjev napetosti, deformacij in hitrosti deformacij. Določevanje sil, momentov in energije, potrebne za preoblikovanje, je z MKE natančnejše kot pri klasičnih metodah. V mnogih primerih nas zanima tok materiala. Pri kovaško-valjavski liniji, na primer, je glavna naloga kovaškega stroja zapiranje notranjih napak v materialu. Simulacija zapiranja notranjih poroznosti v materialu z MKE je grafično prikazana na sliki 10.
RESULTS
Figures 7, 8 and 9 show the distribution of the incre-ment of the equivalent plastic strain developed at three positions of the die during one stroke of the forging ma-chine. The distribution and history of the other ther-momechanical parameters can be similarly represented. These parameters are temperature and ali components and equivalent values of the stress, strain and strain rate tensors. Calculation of forces and energy consumption during the forming process by FEM is more accurate than by classical methods. In many cases the material flovv is of great interest. In the forging-rolling production line, for example, the main purpose of the forging ma-chine is to close the internal rupture. Modelling of the closure of an internal rupture in a continuous čast billet by the radial-forging process is illustrated in Fig. 10.
EOUFVALtNl PLAStlC STRAIN INCREUENT a! »=0 5 <
C
■
RADIAL FORGING PROCESS SHIUIAT/ON
	-1 i—	—-
EOUrttLENT PLJSTIC STRAIN INCREMENT »t »=0 75 »rad		
		
) l *		II
	_1 _	min.- 0.00 /(■
RADIAL FORGING PROCESS SIMULATION	JO	'»i*mc
Slike 7, 8 in 9:
Prirastek plastičnih deformacij v preoblikovalni coni med udarcem kladiva.
Fig. 7, 8 and 9:
Increment of the equivalent plastic strain developed at three positions of the die during one stroke of the forging machine.
Slika 10:
Modeliranje zapiranja notranjih odprtin v konti-liti gredici z radialnim kovanjem.
Fig. 10:
Modelling the closure of an internal rupture in a continuous čast billet by the radial-forging process.
7. ZAKLJUČEK
MK.E je pogosto uporabljena metoda za analizo preoblikovalnih procesov. V primerjavi s klasičnimi metodami ima naslednje prednosti: z MKE lahko rešujemo primere z zahtevno geometrijo preoblikovanca; problem lahko rešujemo z različnimi materialnimi modeli; možno je obravnavanje nestacionarnih napetostno-de-formacijskih in temperaturnih polj. V prihodnje pričakujemo vgrajevanje novih spoznanj s področja fizikalne metalurgije v numerične modele. Prvi koraki v tej smeri so bili že storjeni.18
6. CONCLUSIONS
The FEM is now widely used for the analysis of metal forming processes. In comparison with classical meth-ods the FEM has certain advantages: various complex shapes can be considered, it enables implementation of diferent material models and treatment of transient stress-strain and temperature fields. In the future the in-clusion of the metalurgical development in the numerical modelling of hot vvorking processes is expected. The first steps tovvard this goal have already been made.17'18
LITERATURA/REFERENCES
1.	Sellars C. M.: The Physical Metallurgy of Hot Working The VVorking and Forming Processes (edited by Sellars C. M. and Davies G. J.) The Metals Society, London, U. K., (1979)
2.	Mahrenholtz D. and Dung N. L.: Mathematical Mode/ing of Metal Forming Processes by Numerical Methods. Advanced Technology of Plasticity 1987 (edited by K. Lange),
Vol. 1, pp. 3—10, Springer-Verlag, Berlin Heidelberg New York Tokyo, (1987)
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