Nova metoda za doloŁevanje krivulje mejnih deformacij v

Strojni�ki vestnik - Journal of Mechanical Engineering 51(2005)6, 330-345

330 Petek A. - Pepelnjak T. - Kuzman K.

Nova metoda za doloèevanje krivulje mejnih deformacijv digitalnem okolju

An Improved Method for Determining a Forming Limit Diagramin the Digital Environment

Ale� Petek - Toma� Pepelnjak - Karl Kuzman

Deformacijske meje ploèevine, opredeljene z lokalizacijo in trganjem materiala, so pomembenparameter pri analizi preoblikovalnih postopkov. Meje dopustnih deformacij ploèevine pri razliènihdeformacijskih stanjih najbolje prika�emo v diagramu mejnih deformacij. Za preizkusno doloèitev diagramaso potrebni obse�ni in dragi preizkusi. Alternativna metoda za doloèitev krivulje mejnih deformacij (KMD)je analiza trganja materiala z numeriènimi simulacijami.

V prispevku je predstavljena metoda doloèevanja krivulje mejnih deformacij za celotno podroèje.Z uporabo programskega paketa ABAQUS je bila izvedena simulacija MKE po Marciniaku preizkusavroèe cinkane jeklene ploèevine. Opisan je kriterij za doloèitev KMD z numerièno simulacijo in na koncu �eprimerjava numerièno in preizkusno dobljene KMD.© 2005 Strojni�ki vestnik. Vse pravice pridr�ane.(Kljuène besede: preoblikovanje ploèevine, krivulje mejnih deformacij, metode konènih elementov, simuliranjenumerièno)

The deformation limits of sheet metals, which are determined with localization and fracture, repre-sent an important parameter for the analyses of sheet-metal forming processes. The forming limits of sheetmetal are represented by a forming limit diagram (FLD), which shows the various deformation states. Foran experimental determination of a FLD extensive and expensive tests are necessary. An alternative methodfor determining the forming limit curve (FLC) is an analysis performed using numerical simulations.

This paper introduces a method for determining the forming limit curve for the whole range of theFLD for sheet metal. A simulation of the Marciniak test with the finite element method (FEM) was performedfor hot-galvanized steel using the ABAQUS program. The criterion for determining the FLD with a numeri-cal simulation is presented. Finally, the numerically obtained forming limit curve is compared with anexperimental curve.© 2005 Journal of Mechanical Engineering. All rights reserved.(Keywords: sheet metal forming, forming limit curves, finite element methods, numerical simulations)

Strojni�ki vestnik - Journal of Mechanical Engineering 51(2005)6, 330-345UDK - UDC 621.98.04:539.375Pregledni znanstveni èlanek - Review scientific paper (1.02)

0 UVOD

Krivulja mejnih deformacij (KMD)ploèevine predstavlja mejo, do katere lahko doloèenmaterial preoblikujemo � deformiramo, ne da bi pritem pri�lo do poru�itve. Opredelimo jo v odvisnostiod dveh glavnih plastiènih deformacij j

1 (veèja na

ravnini ploèevine) in j2 (manj�a na ravnini ploèevine),

kakor prikazuje slika 1.V industrijski praksi nas velikokrat zanima,

kako bo potekal preoblikovalni postopek, kje sokritièna podroèja � mo�nost poru�itev in napak.

0 INTRODUCTION

The forming limit curve (FLC) for sheetmetal indicates the point to which a material can beformed before cracks occur on the specimen. Thecurve is defined as a correlation between the firstprincipal strain j

1, which is major in the plane of the

sheet metal, and the second principal strain j2, which

is minor in the plane of the sheet metal (Figure 1).In industrial practice it is often important

how the forming process is performed; it is neces-sary to define where the critical areas of necking and


331Nova metoda za doloèanje krivulje - An Improved Method for Determining

Zaradi prihranka èasa in stro�kov je nujno potrebnoanalizirati tehnologijo preoblikovanja �e predenizdelamo orodja in izvedemo preizkuse. V ta namense v zadnjem èasu vse bolj uporabljajo sodobnenumeriène simulacije, ki v t. i. �raèunalni�kem okolju�prika�ejo potek preoblikovalnega postopka. Èeugotovimo, do katere meje lahko preoblikujemodoloèen izdelek, lahko postopek optimiramo, s temprihranimo èas, zni�amo stro�ke in izbolj�amokakovost.

Zaèetek analize mejne deformacije izhaja izleta 1940. Prvi diagram, ki je bil podoben tipiènemudiagramu mejnih deformacij (DMD), je objavilGansamer leta 1946, kot opisujeta Geiger in Merklein[1]. Leta 1965 je Keeler [2] razvil zasnovo DMD, kakorje poznana dandanes. Keeler je s preizkusi ugotovil,da je krivuljo mejne deformljivosti za ploèevinomogoèe prikazati v koordinatnem sistemu dvehglavnih deformacij, vendar le za podroèje j

2>0. Ob

koncu 60. let je to zamisel raz�iril Goodwin [3], ki jedopolnil diagram za obmoèje j

2<0. Zadnjih 40 let je

diagram mejnih deformacij vzbudil pomemben vtisna raziskovalne ustanove in industrijo predvsem privpra�anju, kako doloèiti najveèje deformacije prikaterih �e ne bo pri�lo do poru�itve materiala medpreoblikovanjem. Zaradi poveèane uporabenumeriènih simulacij ploèevinskih postopkov, ki soposledica razvoja zmogljivej�ih raèunalni�kihsistemov, je bil razvoj na podroèju KMD �e boljpospe�en.

Razvoj doloèevanja KMD je usmerjen v triglavna podroèja: teoretièno, preizkusno in numeriènodoloèevanje.

fracture are. The forming technology can be ana-lysed before the tool is manufactured, which leadsto savings in costs and time. For this purpose nu-merical simulations have recently been used to showthe course of the forming process in a �digital envi-ronment�. If the forming limit for a particular productis known the process can be optimized. In this waytime is saved, costs are reduced and the quality ofproducts is improved.

The analysis of forming limits began in the1940s. The first diagram, which was similar to the typi-cal FLD, was published by Gansamer in 1946, as wasdescribed by Geiger and Merklein [1]. In 1965 Keeler[2] developed the concept of the nowadays knownFLD. By using experiments Keeler realised that it waspossible to show a FLC for a sheet metal in a coordi-nate system of two main strains, but only for the right-hand side of the contemporary known FLD (j

2>0).

This idea was extended by Goodwin [3] at the end ofthe 1960s when the diagram was completed for left-hand side with deformations of j

2<0. Over the past 40

years, the concept of the forming limit diagram hascreated a significant impact in both academia and in-dustry on how we determine the maximum deforma-tion that a material can withstand, without necking ortearing, during a sheet metal process. Since the en-hanced use of numerical simulations in the sheet metalforming stimulated due to the immense increase incomputer power, the necessity on research work inthe field of FLD has also intensified.

The development of determining FLDs isoriented in three main fields: theoretical, experimentaland numerical determination.

Sl. 1. Diagram mejnih deformacij (Ha�ek) [1]Fig. 1. Forming limit diagram (according to Ha�ek) [1]



Po razvoju zamisli diagrama (Keeler [2],Goodwin [3]) se je raziskovanje preoblikovalnostimateriala usmerilo predvsem na razvoj matematiènihmodelov za teoretièno doloèevanje krivulje mejnihdeformacij. Prva, ki sta predlagala kriterij lokalizacijena tanki ploèevini za ravninsko napetostno stanje,sta bila Hill [4] in Swift [5]. Njuna analiza napovelokalizirano plastièno deformacijo v podroèju j

2<0,

pri èemer je model temeljil na homogenih lastnostihploèevine. Prvi matematièni model za teoretiènodoloèevanje krivulje, ki temelji na nehomogeniploèevini z vidika geometrije in sestave, sta razvilaMarciniak in Kuzinsky, in je danes bolje poznankot model M-K [6]. Eden najbolj�ih teoretiènihmodelov, ki se zelo dobro ujema s preizkusnimipodatki, je na konferenci IDDRG predstavilCayssials leta 1998 [7]. Zaradi velike natanènostise Cayssials model uporablja v sodobnihraèunalni�kih programih za analizo preoblikovanjaploèevine v digitalnem okolju.

Zgoraj omenjeni teoretièni modeli so zelozapleteni in zahtevajo temeljito predznanje mehanikein matematike. Izraèunane KMD �al niso vednoprimerljive z preizkusnimi rezultati, zato so v zadnjihletih razvili tudi delno empiriène modele. Prvi avtordelno empiriènega modela je bil Keeler.

Potreba po zmanj�anju obse�negaanalitiènega izraèunavanja in mo�nosti hitregadoloèevanja KMD tudi v industrijskem okolju jeprivedla do prouèevanja natanènosti in uèinkovitostiopredelitve krivulj mejnih deformacij s preizkusi [8].Za doloèitev razliènih deformacijskih stanj so bilemed preoblikovalnim postopkom zahtevane razliènegeometrijske oblike orodja. Predstavljene so bileposebne metode za doloèitev razliènih napetostnihstanj z eno geometrijsko obliko orodja in razliènooblikovanimi preizku�anci.

Prvo �iroko uporabljeno metodo je leta 1971predlagal Nakazima [9], ki je za doloèitev vsehdeformacijskih stanj na DMD uporabljal eno orodje.Uporabljeno je bilo preoblikovalno orodje,sestavljeno iz polkro�nega pestièa, matrice in dr�ala.Za doloèitev razliènih deformacijskih stanj souporabljeni pravokotni preizku�anci razliènih �irin.Slaba stran te metode je merjenje deformacij napolkro�nem preizku�ancu, za kar sta potrebni najmanjdve kameri.

Leta 1973 je Marciniak [10] predstavilpodobno metodo za doloèevanje KMD (sl. 2). Vnasprotju s prej�njo metodo ostaja priMarciniakovem preizkusu analizirano podroèje med

After the evolution of the concept of FLDs(Keeler [2], Goodwin [3]), the research on materialformability was focused mainly on the developmentof mathematical models for theoretical determinationof FLDs. Hill [4] and Swift [5] were the first to proposea general criterion for localized necking in thin sheetsunder a plane stress state. Their analyses predictedlocalized plastic deformation in the negative minorstrain region (j

2<0). The model assumed homogeneity

of the sheet metal. Marciniak and Kuzinsky haveproposed the first realistic mathematical model for atheoretical determination of FLDs taking into accountthe non-homogeneous material behaviour of theanalysed sheet metals from the geometrical andstructural points of view. This model is nowadaysknown as the M-K model [6]. One of the besttheoretical model, which fits to the experimental data,was introduced by Cayssials at the IDDRG conferencein 1998 [7]. Because of its high precision the majorityof computer programs use the Cayssials model forsheet-metal analyses in the digital environment.

The above-mentioned theoretical modelsare rather complex and need a profound knowledgeof the continuum mechanism and mathematics. Theo-retically calculated FLCs are not always in agree-ment with the experimental data. Therefore, somesemi-empirical models have been developed. Thefirst author of a semi-empirical model was Keeler.

The necessity to reduce the extensive ana-lytical calculations and the possibility of a fast de-termination of the FLC in the industrial environmentled to a study of the precision and efficiency of defi-nition of the FLC with experiments [8]. To obtainvarious strain states during the forming process dis-similar tool geometries are required. Special meth-ods for determining different stress states with onlyone tool geometry and various shaped tests are pro-posed.

The first widely applied method was intro-duced by Nakazima in 1971 [9]. He used a single toolto determine all the strain states of the FLD. The toolcomposed of a hemispherical punch, a die and a blankholder. Rectangular test pieces with various widthsfor a definition different deformation states are used.The drawback of this method is the necessity for atleast two cameras for strain measurements on thehemispherical test piece.

A similar method for determining FLCs (Fig-ure 2) was introduced by Marciniak in 1973 [10]. Incontrast to Nakazima method the investigated regionin the Marciniak test remains flat during the experiment.



preoblikovanjem ravno. Merjenje deformacij lahko,zaradi 2D - problema, izvedemo z eno kamero, kar jebistvena prednost te metode. Zaradi mo�nostinastanka poru�itve preizku�anca na polmeru pestièauporabljamo vodilni obroè, ki tak�no trganje prepreèi.Uspe�nost metode je odvisna od natanènegeometrijske oblike vodilnega obroèa in trenja medpreizku�ancem in vodilno ploèevino.

Za analizo deformacij ploèevinepotrebujemo pri obeh metodah na povr�inianaliziranega preizku�anca natisnjeno mre�o.Deformacija mre�e je bila vèasih merjena s posebnimimikroskopi, medtem ko se v novej�em èasuuporabljajo optièni merilni sistemi s CCD kamerami.

Geiger in Merklein sta na vsakoletnemsreèanju CIRP leta 2003 predstavila zanimivo zamiselo povezavi med lokalizacijo in gradientom glavnedeformacije [1] pri Nakazima metodi.

Poleg zgoraj omenjenih metod doloèevanjaKMD je bilo v zadnjih desetih letih predstavljenih �eveliko drugih metod, kakor so enoosni nateznipreizkus, s katerim lahko doloèimo KMD le zapodroèje j

2<0, hidravlièen izboèitveni test, Keeler

test, Hecker test, Ha�ek test in druge [11].V zadnjih letih se zaradi potrebe po

zmanj�anju stro�kov in �tevilu preizkusov zadoloèevanje KMD vse veè uporabljajo simulacijeMKE. Brun [12] predstavlja zamisel o analizilokalizacije z drugim odvodom debeline po èasu zNakazima metodo, medtem ko Ozturk analizira trganjemateriala s kriterijem �ilave poru�itve na istipreizkusni metodi [13].

Podroèje lokalizacije in kasneje zloma napreizku�ancu je z uporabo numeriène simulacije te�ko

Therefore, because this case is a 2D-application, strainscan be measured with only one camera, an importantadvantage of this method. Due to the danger of fracture,which can appear outside the observed specimen area, aguiding blank is used. This prevents such a failure. Theeffectiveness of this method depends on the geometri-cal precision of the guiding blank and the friction be-tween the test piece and the guiding blank.

The deformation analysis in both methodsrequires a grid system printed on sheet-metal sur-face. Originally, the deformed grids were measuredwith special microscopes, but nowadays opticalmeasuring systems with CCD cameras are used.

At the annual CIRP meeting in 2003 an ideaabout the correlation between the materials� neck-ing and the gradient of major strain versus time dur-ing the Nakazima method was presented by Geigerand Merklein [1].

In addition to the above-mentioned meth-ods for determining the FLC, many other methodswere introduced in the past ten years, e.g., the uniaxialtensile test, which defines only the left part of theFLD (j

2<0); the hydraulic bulge test; the Keeler test;

the Hecker test; and the Ha�ek test [11].In recent years demands on cost reduction

and decreased number of tests necessary for defini-tion of the FLC have intensified the use of finite-ele-ment simulations. Brun [12] introduced the idea thatthe onset of necking can be predicted by the secondtime derivative of thinning by the Nakazima method,whereas Ozturk analysed the material failure using theductile fracture criteria applied in the same test [13].

The onset of necking and the subsequentfracture of the specimen are hard to predict with

Sl. 2. Orodje Marciniak metodeFig. 2. Tool for the Marciniak method

PestièPunch

Dr�aloBlank-holder

Vodilna ploèevinaGuiding blank

MatricaDie

PlatinaBlank

Dr�alo



napovedati. V prispevku je predstavljen inovativenpostopek o iskanju kriterija za re�itev zgorajomenjenega problema.

Razlièna deformacijska stanja v ploèevinidose�emo z razliènimi preizkusi. V praksi �elimouporabiti èim manj preizkusnih orodij, zato �elimoKMD izdelati s preizkusi, pri katerih se spreminjaoblika preizku�anca in ne oblika orodja. Izomenjenega razloga smo tako pri preizkusnem, kakorpri numeriènem doloèanju KMD uporabiliMarciniakov izboèitveni preizkus.

Tipièni popis diagrama mejnih deformacij seizvede s �tirimi razliènimi deformacijskimi stanji(globoki vlek, natezni preizkus, enoosno deformacijskostanje in enakomerno izboèevanje), pri katerih v daniploèevini povzroèimo lokalno stisnjenje in natoporu�itev materiala. Pojav zaèetka lokalnega stisnjenjase �teje kot indikator poru�itve preoblikovalnegapostopka za ploèevino. V tem trenutku se na kritiènemobmoèju preizku�anca pojavijo deformacije, kidefinirajo toèke krivulje mejnih deformacij na diagramuj

1 v odvisnosti od j

2. Deformacijo na mestu poru�itve

pri preizku�anju izmerimo z uporabo grafometrièneanalize ([8] in [14]). Analiza temelji na ugotavljanjuvelikosti in smeri deformacij na posameznem obmoèjuploèevine na osnovi spremembe koordinatne merilnemre�e, ki jo pred preoblikovanjem nanesemo napreizku�anec. Pri numeriènem doloèanju padeformacijo na mestu poru�itve doloèimo z uporaboalgoritma, opisanega v tem prispevku.

1 OPIS NUMERIÈNEGAMODELA

Model za simulacijo Marciniakovegapreizkusa je sestavljen iz togih delov orodja, kakor sovaljasto oblikovan pestiè z ravnim poglobljenim èelom,zgornja plo�èa (dr�alo) in spodaj odprta matrica zzavorno letvijo in deformljivih delov, kakr�na stapreizku�anec in vodilni obroè (sl. 3). Pri Marciniakovimetodi uporabljamo �e omenjen vodilni obroè, prikazanna sliki 3, ki prepreèuje nastanek najveèjih deformacijpreizku�anca v bli�ini polmera pestièa. Ta obroè sedeformira skupaj s preizku�ancem, pri èemer premestideformacije preizku�anca iz polmera pestièa na ravnopovr�ino pod pestièem. Geometrijska oblika orodja jebila izbrana glede na objave tujih raziskav.

Marciniakov preizkus zahteva uporaborazliènih geometrijskih oblik preizku�ancev zadoloèitev razliènih deformacijskih stanj in napetosti.Simulacijski postopek je potekal na sedmih

numerical simulations. This paper presents an inno-vative approach to finding alternative criteria to solvethe above-described problem.

Different strain states in sheet metal arereached with various tests. In practice it is desirableto minimise the number of used forming tools. Wewant to define the FLC using tests where thespecimens� shapes change and the tool geometryremains unchanged. For this reason the Marciniaktest was selected for the experimental as well as forthe numerical determination of the FLC.

A typical description of the FLD is executedwith four characteristic deformation states (deepdrawing, uniaxial tension, uniform tension and biax-ial balanced stretch forming), by which the localiza-tion and the subsequent material fracture in the sheetmetal is caused. The onset of necking is consideredas a fracture indicator of the forming process forsheet metal. At this point limit strains on the criticalarea of the specimen appear, which define the pointsfor the limit curve on the diagram j

1 versus j

2. De-

formation, where the fracture occurs, is measuredusing graphometric analyses ([8] and [14]). Thisanalysis is based on size and direction investigationof the major strains of the particular sheet metal areaby changing the coordinate measure grid, which isprinted on the specimens before the forming proc-ess. However, with a numerical simulation the frac-ture strain is determined using the methodologydescribed in this paper.

1 DESCRIPTION OF THE FINITE-ELEMENTMODEL

The finite-element model for the Marciniaktest consists of rigid tool parts: a cylindrical punchwith a flat bottom, a blank holder and a die withdrawbead and deformable parts, like the specimenand the guiding blank (Figure 3). The guiding blankused in the Marciniak method, as shown in Figure 3,prevents the appearance of maximal strains in thevicinity of the punch radius. This blank is deformedtogether with the specimen and helps to repositionthe critical strains from the punch radius to the flatsurface under the punch. The tool geometry waschosen after consulting the publications frominternational research work.

The Marciniak test requires the applicationof various specimens� geometries for a determinationof the different strain states and stresses. Thesimulation process was achieved with seven different



preizku�ancih razliènih oblik (sl. 4). Prvi preizku�anecima obliko kroga premera 200 mm (rondela), preostalih�est pa ima obliko traku (platina), ki je bil izrezan izkroga s premerom 200 mm, s �irinami 165 mm, 150mm, 135 mm, 130 mm, 95 mm in 60 mm. Vsak odvzorcev pomeni na diagramu mejnih deformacij enospecifièno deformacijsko pot. Za numeriènosimuliranje omenjenega preoblikovalnega postopkasmo uporabili programski paket ABAQUS, ki temeljina metodi konènih elementov (MKE). Izbrali smo 3Dlupinske elemente z veè integracijskimi toèkami podebelini lupine. Zaradi velikega �tevila objektovsimulacije, ki so v medsebojnem kontaktu, smo se vizogib kontaktnim problemom pri implicitnem naèinuraèunanja odloèili za eksplicitni naèin raèunanja.

Mehanske lastnosti preizku�anca (vroèecinkana jeklena ploèevina), ki jih potrebujemo zapopis elasto�plastiènega obna�anja materiala, smodobili z neprekinjenim enoosnim nateznim preizkusomv Laboratoriju za preoblikovanje na Fakulteti za

specimen geometries (Figure 4). The first specimenis a blank with a 200 mm diameter; the other six arestrip-shaped cut from the same blank with a diameterof 200 mm. The widths of the strips are 165 mm, 150mm 135 mm, 130 mm, 95 mm and 60 mm. Each samplecorresponds to a specific strain path on the FLD.For the simulations of the Marciniak method, thecommercially available finite-element programABAQUS was used. Three-dimensional shellelements with five integration points across thethickness were selected. Due to the large number ofsimulation objects in contact, solver convergenceproblems can occur during the simulation. An explicitsolver was selected to alleviate this problem.

The material properties of hot-galvanizedlow-carbon steel necessary to define the elasto-plastic material behaviour were obtained using a uni-axial tensile test in the Forming Laboratory of theFaculty of Mechanical Engineering in Ljubljana ([15]and [16]). The stress-strain curve was approximated

Sl. 3. Objekti simulacije Marciniak preizkusaFig. 3. Finite element model of Marciniak test

Sl. 4. Razliène oblike preizku�ancevFig. 4. Different shapes of test pieces

Matrica Die

Pestic Punch

Dr�alo Blank-holder

Vodilna ploèevina Guiding blank

Rondela Blank

O 200 mm 95 mm130 mm135 mm150 mm165 mm 60 mm

Pestiè



strojni�tvo v Ljubljani ([15] in [16]). Preraèun pribli�kakrivulje plastiènosti smo izvedli po Hollomonovempotenènem zakonu. Mehanske lastnosti so prikazanev preglednici 1.

Material je bil popisan z elasto�plastiènimsnovnim zakonom. Upo�tevali smo anizotropnoobna�anje materiala, ki smo jo popisali s Hill-ovimkvadratiènim kriterijem teèenja. Menimo, da modelMKE ni odvisen od hitrosti preoblikovanja.Simulacijski postopek je bil razdeljen na dva koraka.V prvem koraku se dr�alo giblje v smeri z in pritisnevodilno ploèevino ter preizku�anec ob matrico. Zobmatrice deformira material in prepreèuje drsenjepreoblikovanca med dr�alom in matrico. Matrica inpestiè ostajata v tem koraku nepremièna. Sila dr�alaje med celotnim postopkom simuliranja pribli�no 150kN. V drugem koraku preoblikovalnega postopka sepestiè giblje v smeri z, dokler ne dose�e zahtevaneoddaljenosti.

Trenje med dotikalnimi povr�inamiposameznih delov Marciniakovega modela jedefinirano s Coulombovim zakonom. Vrednostikoeficientov trenja m med posameznimi telesisimulacije v dotiku, dobljene iz preizkusov, so:

by the Hollomon potential law. The materialproperties are shown in Table 1.

In the FE simulations the elasto�plasticmaterial behaviour was modelled. The anisotropicmaterial properties were considered, which weredescribed with the Hill quadratic yield criterion. Themodel was assumed to be rate independent. Thesimulation processes were performed in two steps.In the first step the blank holder moves down in the�z�-direction and presses the guiding blank and thesheet-metal specimen onto the die. The die drawbeaddeforms the material and prevents the specimensliding between the blank holder and the die. Thecylindrical punch and the die remain fixed in thisstep. The blank-holder force is approximately 150kN during the whole simulation process. In thesecond simulation step the punch moves with aconstant speed in �z�-direction until the prescribeddisplacements are achieved.

The contact interaction between thesurfaces in the FE model is defined with the Coulombfriction law. The friction coefficients, m, among thesimulation objects in contact have the followingvalues, obtained from the experiments:

Preglednica 1: Mehanske lastnosti vroèe cinkane jeklene ploèevineTable 1: Material property of hot galvanized steel

C = 719,2 MPa E = 210 GPa n = 0,153 r = 7850 kg/m3

r0 = 0,948 n = 0,3 r45 = 0,793 r90 = 1,003

t0 = 0,64 mm

C�..konstanta utrjevanja materiala [MPa] n�..eksponent utrjevanja [1] r�.. koeficient normalne anizotropije [1] E�..modul elastiènosti [MPa] r�..gostota [kg/m3] n�..Poissonovo �tevilo [1] t0�..zaèetna debelina preizku�anca

C�..strength coefficient [MPa] n�..strain hardening coefficient [1] r�...material anisotropy [1] E�..Young�s modulus of elasticity [MPa] r�..density [kg/m3]

n�..Poisson�s ratio [1] t0�..initial specimen thickness

pri èemer so: where the parameters are:

- pestiè � vodilna ploèevina: m = 0,08, - dr�alo � vodilna ploèevina: m = 0,25, - vodilna ploèevina � preizku�anec: m = 0,3, - preizku�anec � zob matrice: m = 0,25, - preizku�anec � polmer matrice: m = 0,08.

- Punch � guiding blank: m = 0.08, - Blank holder � guiding blank: m = 0.25, - Guiding blank � blank: m = 0.3, - Blank � die drawbead: m = 0.25, - Blank � die radius: m = 0.08.



Mre�a in robni pogoji MKEMarciniakovega modela so prikazani na sliki 5.Posebno pozornost smo posvetili mre�ipreizku�anca. Zaradi prièakovanega nastankalokalizacije oziroma zloma preizku�anca vpodroèju pod pestièem, to podroèje omre�imo nazelo majhne kvadratne elemente mre�e velikosti1,5 x 1,5 mm2. S tem zmanj�amo vpliv velikosti inoblike mre�e na natanènost numerièno dobljeneKMD [17], medtem ko se èas raèunanjanumeriène simulacije poveèa. Velikostpravokotnega podroèja je pri preizku�ancih, �ir�ihod 120 mm, veliko 120 x 120 mm2. Pri o�jihpreizku�ancih pa se to podroèje primerno zo�i.

2 OPIS KRITERIJA ZADOLOÈEVANJE KMD

Pri analizi KMD preiskujemo nastaneklokalizacije oziroma poru�itve na preizku�ancu medpostopkom preoblikovanja. Doloèiti je potrebnoobmoèje ter èas nastanka lokalizacije in kasnejeporu�itve. Zamisel o kriteriju za doloèevanje KMD znumerièno simulacijo smo dobili iz teoretiènegamodela Marciniaka�Kuckzinskega (M-K), ki temeljina doloèevanju lokalizacije na podroèju vnesenegeometrijske napake. Kratek opis modela M-K jepredstavljen v naslednjem poglavju. Prav tako je bilakoristno uporabljena Geigerjeva gradientna metoda

The mesh and boundary condition of theMarciniak finite-element method are presented as onequarter of the entire model - Figure 5. Special attentionwas given to the specimen�s mesh. Due to theexpectation of localized necking and sample fractureunder the punch this area was meshed with a verysmall squared mesh with a size of 1.5 x 1.5 mm2.Implementing a so small grid size, the influence of themesh size and its shape on the accuracy of thenumerically obtained FLC [17] decreases; however, thecalculation time of the numerical simulation increases.Specimens wider than 120 mm have a mapped mesharea of 120 x 120 mm2. When using narrow specimensthis area is correspondingly narrower.

2 METHODOLOGY DESCRIPTION FOR DETER-MINATION OF THE FLC

The onset of necking or fracture of theformed specimen is investigated when the numeri-cal FLC is analysed. The time and area of the onsetof necking and subsequent fracture is defined. Sev-eral ideas about the criterion for the determinationof the numerical FLC are obtained from the theoreti-cal Marciniak�Kuckzinsky (M-K) model. The modeldescribes the localization in the area with the in-serted geometrical defect. A brief description of theM-K model is presented in the following section.Geiger�s and Merklein�s gradient method [1] was also

Sl. 5. Model konènih elementov (1/4)Fig. 5. Finite element model (1/4)

v=5 m /s

F=150 kN

M atricaD ie

Dr•a loBlank - ho lder

PlatinaBlank

Vodilna p lo evinaG uid ing b lank

è

PestiPunch

è

�



[1]. S preizkusnim postopkom je ugotovil, da se gra-dient deformacije med preoblikovalnim postopkomnaglo spremeni v trenutku nastanka lokalizacije.Podobno zamisel o uporabi drugega odvodadeformacije debeline po èasu, kot kriterijlokalizacije, je v raèunalni�kem okolju predstavilBrun [12]. Metoda napove, kdaj se pojavi lokalizacijain ni primerna za analiziranje, kje in kako se je le tapojavila.

2.1 Kratek opis teoretiènega modela Marciniaka �Kuckzinskega

Z uporabo modela M-K analiziramoplastièno nestabilnost v materialu. Pri temupo�tevamo plastièni model z izotropnimutrjevanjem materiala in ravninsko napetostnostanje.

Model temelji na poveèevanju zaèetnenapake v obliki ozkega utora, nagnjenega podkotom y

0 glede na os y (sl. 6). Zaèetna velikost

geometrijske napake je karakterizirana z razmerjem

0 0b ae e , kjer je 0

ae zaèetna debelina podroèja in 0be

debelina utora. Osi x, y in z ustrezajo vzdol�ni,preèni in pravokotni smeri glede na smer valjanjaploèevine, medtem ko 1 in 2 predstavljata glavniosi napetosti in deformacij v homogenempodroèju.

Smeri na utoru so predstavljene z osmi n, t,z, kjer je �t� vzdol�na os. Obe podroèji materiala staizpostavljeni plastièni deformaciji, pri èemerupo�tevamo stalno raztegovanje homogenega dela.Plastièna deformacija se pri obremenitvi pojavi vobeh podroèjih. Njena velikost pa je odvisna odpreènega prereza podroèja. Ob dosegu kritiène

found to be useful. They established with their ex-perimental work that the strain gradient during theforming process changes rapidly at the instant whenthe localization occurs. A similar idea, analysed in thedigital environment, about the usage of the secondtemporal derivation of the thickness strain as locali-zation criteria is presented by Brun [12]. The methodpredicts when the onset of necking occurs but it isnot suitable for analysing where and how it appears.

2.1 Brief description of theoretical Marciniak�Kuckzinsky model

The simulation of the plastic instability isperformed using the M-K analysis. The rigidplasticity, the plane stress condition and the isotropicwork hardening of the material are assumed.

The model is based on the growth of aninitial defect in the form of a narrow band inclined atan angle y

0 with respect to the principle axis. The

initial value of the geometrical defect is characterizedby the ratio 0 0

b ae e , where 0ae and 0

be are the initialthicknesses in the homogeneous region and thegroove, respectively. The x, y and z axes correspondto the rolling, transverse and normal directions ofthe sheet, whereas the 1 and 2 axes represent theprincipal stress and strain directions in thehomogeneous region, respectively.

The set of axes bound to the groove is rep-resented by the n, t and z axes, where �t� is orientedin the longitudinal direction. This two-zone materialis subjected to plastic deformation by applying aconstant incremental stretching of the homogene-ous part. The plastic flow occurs in both regions,but the evolution of the strain rates is different in

Sl. 6. Zaèetna napaka M-K modela [18]Fig. 6. Initial defect of the M-K model [18]



deformacije v utoru se v njem pojavi lokalizacija. Pritem je dose�ena mejna deformacija ploèevine. Za la�joizpeljavo ravnote�nih enaèb modela M-K sepredpostavi, da se deformacija (j

1) pojavi v smeri

osi x.Natanèen opis teoretiène analize M-K,

shematsko prikazane na sliki 6, lahko zasledimo vrazliènih virih ([6],[18] do [20]).

2.2 Opis numeriènega kriterija za doloèitevKMD

Pri iskanju kriterija za doloèitev numeriènodobljene krivulje mejnih deformacij smo najprejanalizirali diagram preoblikovalne sile, ki ga dobimoz numerièno simulacijo za vsako izmero vzorcaposebej.

V splo�nem prikazuje preoblikovalna silanara�èajoèo pot do dosega njene najveèjevrednosti in nenadno zmanj�anje ob pojavuporu�itve ploèevine, kakor se to zgodi pripreizkusu. Pri Marciniakovi metodi uporabljamovodilno ploèevino, ki se zaradi bolj�ihpreoblikovalnih lastnosti in veèje debeline poru�ikasneje kakor preizku�anec. Kasnej�a poru�itevvodilne plo�èe pri simulaciji z MKE zadr�ujenenadno zmanj�anje sile, ki se pojavi pri poru�itvipreizku�anca. Poru�itev skupine vzorcev, kipopisujejo levo stran KMD, se pojavi po celotnidrsni ravnini, pri èemer se le-ti razpolovijo. Taceloten zlom preizku�ancev vpliva na pojavnenadnega zmanj�anja sile preoblikovanja kljubuporabi vodilne ploèevine. Zato je mogoèe pojavnenadnega zmanj�anja sile pri Marciniakovimetodi opazovati le pri preizku�ancih, s katerimipopi�emo levo stran KMD.

Èe vri�emo deformacijsko stanjeposameznih preizku�ancev v trenutku, ko sepreoblikovalna sila zmanj�a, in èe upo�tevamo samonajvi�je toèke posameznih preizku�ancev, jenatanènost narisane KMD zelo majhna. Iz tega lahkosklepamo, da je preoblikovalna sila preveè splo�enparameter in zato ni najbolj primerna za opredelitevmeje poru�itve ploèevine.

Za doloèitev pojava lokalnega stisnjenjapotrebujemo manj splo�en parameter, kakor jepreoblikovalna sila, ki mora biti povezan sposameznim elementom oz. vozli�èem. Iz naravelokalnega stisnjenja debeline ploèevine ter obupo�tevanju analize M-K, bi lahko za parameter izbraliplastièno deformacijo debeline najtanj�ega vozli�èa.

two zones. When the flow localization occurs in thegroove at a critical strain in the homogeneous re-gion, the limiting strain of the sheet metal is reached.Furthermore, the major strain is assumed to occuralong the x axis.

A detailed description of the theoreticalM-K analysis, schematically illustrated in Figure 6,can be found in several publications ([6],[18] to [20]).

2.2 Description of the numerical criterion for thedetermination of the FLC

The first attempt made by an investigationof the criterion for the determination of thenumerically achieved FLC was the analysis of theforming force versus time for all the simulatedspecimen dimensions.

The stamping force obtained by the FEMshows an increasing ranking until its maximum value isreached and abruptly decreases as soon as failureappears on the sheet, as also happens with the experiment.The guiding blank, which has better forming propertiesand/or greater thickness, is used by the Marciniak methodand fails later than the specimen. The subsequentfracture of the guiding blank by the FEM simulationdelays the sudden decrease of the forming forceappearing as a result of material fracture. The fracture ofall the specimens describing the left part of the forminglimit diagram (j

2<0) is appears on the whole slide plane.

In these cases the samples are divided into twoapproximately equal parts. This specimens rupture resultsin a suddenly decrease in the forming force, despite theuse of a guiding blank. Therefore, the appearance of anabrupt decrease of the forming force during the Marciniakmethod can only be seen for samples defining the left-hand side of the FLD.

If the strain state of the particular specimensat the moment when the forming force drops down isplotted, and only the highest points of severalspecimens are considered, the drawn FLC is notaccurate enough. It can be concluded that the formingforce is a too general parameter and is thereforeprobably not the most suitable one to describe thedeformation behaviour of the sheet metal.

For the determination of the phenomenonof localized necking, a criterion related to the singlenode or element is needed. This is a less generalparameter than the forming force. With considerationof the M-K analysis and the nature of the localcontraction of sheet thickness, the plastic thicknessstrain of the thinnest node can be chosen. When



Ob pojavu lokalnega stisnjenja moramo zaznati ostrospremembo v deformacijskem obna�anju izbranegavozli�èa. Na krivulji, ki prikazuje logaritemskodeformacijo v smeri debeline ploèevine j

s v

odvisnosti od èasa (sl. 7), vidimo koleno, ki poudarjaskokovito spremembo deformacije.

Èe predvidevamo, da je zaèetek lokalnekontrakcije povezan z znaèilno spremembo hitrostideformacije debeline v odvisnosti od èasa, potemima drugi èasovni odvod deformacije ( )s tj&& v temtrenutku najveèji ekstrem funkcije. Slika 8 prikazuje

necking appears, a sharp change in the strainbehaviour of the selected node should be noted. Onthe curve which shows the true principal strain inthe sheet thickness direction j

s versus time (Figure

7), a knee can be observed. This underlines a biggervariation of strain against time.

If we consider that the onset of neckingcan be connected with a significant variation in thestrain thickness velocity versus time, then thesecond temporal derivation of strain ( )s tj&& has atthis moment maximal extreme function. The first and

Sl. 7. Tanj�anje najtanj�ega vozli�èa v odvisnosti od èasa (platina 95 mm)Fig.7. Thinning as a function of time for the critical node (platinum 95 mm)

Sl. 8. Zaèetek lokalizacije in njena lega na preizku�ancuFig. 8. Onset of necking and its location on the analysed specimen



prvi in drugi odvod deformacije debeline v odvisnostiod èasa. Drugi odvod prikazuje lokaliziran vrh v èasu.V tem trenutku lahko prièakujemo prièetek lokalnegastisnjenja na vzorcu.

Ovrednotenje stanj glavnih deformacij (j1

in j2) razliènih geometrijskih preizku�ancev,

analiziranih glede na ekstreme funkcije ( )s tj&& kritiènihvozli�è posameznega preizku�anca, daje toèke zadoloèitev KMD z uporabo numeriènega postopka.

2.3 Opis metodologije

Pri prouèevanju kriterijev za doloèitev KMDz uporabo numeriènih simulacij se je kot najbolj�aizkazala analiza debelinske deformacije. Pri vrednotenjunumerièno opredeljene KMD smo uporabili sedemrazliènih geometrijskih oblik preizku�ancev.

Uporabljena metodologija pri doloèanjuKMD s pomoèjo numeriènih simulacij je povzeta vnaslednjih sedmih korakih:1 � Iskanje najtanj�ega vozli�èa. Na analiziranemmodelu MKE i�èemo najtanj�a vozli�èa preizku�ancaza celoten èas preoblikovanja, ki ga diskretiziramona èasovne korake t

0.

2 � Doloèanje deformacije debeline. Najtanj�avozli�èa vseh èasovnih korakov so oznaèena invnesena v program za MKE analize ABAQUS.Omenjenim vozli�èem zapi�emo pripadajoèedeformacije debeline med celotnim postopkompreoblikovanja.3 � Izraèun prvega in drugega èasovnega odvodadeformacije debeline. Predvidevamo, da je zaèeteklokalne kontrakcije povezan z znaèilnim poveèanjemdeformacije debeline v odvisnosti od èasa. Skokovitospremembo j

s najbolje prikazuje drugi èasovni odvod

deformacije, ki ga izraèunamo in analiziramo za vsakonajtanj�e vozli�èe. Drugi odvod deformacijeopredelimo kot:

4 � Oznaèevanje najveèje vrednosti drugegaèasovnega odvoda za posamezno vozli�èe. Izraèunanasta prvi in drugi èasovni odvod plastiène deformacije.Oba imata vrh strogo lokaliziran v èasu, okolikaterega se pojavi tudi koleno deformacijske krivulje.Vrh drugega odvoda, ki pomeni precej�no razliko vhitrosti deformacije, prika�e èas, pri katerem naj bi sepojavila numerièno dobljena lokalna kontrakcija.5 � Izbiranje najveèje vrednosti drugega èasovnegaodvoda med vsemi vozli�èi. Napi�emo algoritem, ki

second derivatives of thickness strain versus timeare shown in Figure 8. The later one represents alocalized peak in time. At this moment we can expectthe onset of necking in the specimen.

An evaluation of the main strain states (j1

and j2) for different specimen geometries, analysed

according to the function extremes ( )s tj&& of the criticalnodes on a particular specimen, gives the points for adetermination of the FLD by a numerical approach.

2.3 Methodology description

The thickness strain proved to be the bestparameter for an analysis of the numericallyobtained FLC. Seven different specimen geometrieswere used by evaluating the numerically definedFLC.

The methodology used for a determinationof the FLC with numerical simulations can besummarised in the following seven steps:1 � Searching the most thinned node. On theanalysed FEM model the most thinned nodes of thespecimen for the whole forming time, which isseparated into time intervals of t

0, are searched.

2 � Defining the thickness strain. The mostthinned nodes of all the time increments areindicated and entered in program for the FEManalyses (ABAQUS). For that, nodescorresponding to the thickness strain during thewhole forming process are recorded.3 � Calculation of the first and second temporalderivatives of thickness strain. It is supposed thatthe onset of necking is related with a typical increaseof thickness strain versus time. A sharp variation ofj

s is best represented by a second temporal

derivative, which is calculated and analysed for eachof the most thinned nodes. The second strainderivative is defined as:

(1).

4 � Indicating the maximal value of the second temporalderivative for an individual node. The first and secondtemporal derivatives of plastic strain are calculated. Bothhave a peak that is rigorously localized in time, aroundwhich the knee of the strain curve appears. The peak ofthe second derivative, which acts as a large variation inthe deformation velocity, represents the time in whichthe numerically defined necking is supposed to appear.5 � Selecting the largest value of second temporalderivative among all the nodes. The algorithm was

2

2s

s

d

dt

jj =&&



primerja èase nastankov najveèjih vrednosti drugegaodvoda posameznih vozli�è preizku�anca in zapi�eèas nastanka ter �tevilko vozli�èa, pri katerem se jenajveèja vrednost drugega odvoda najhitrejepojavila.6 � Zapis glavnih deformacij v preèni (j

1) in vzdol�ni

(j2) smeri vozli�èa. Zapi�emo glavni deformaciji v

preèni in vzdol�ni smeri vozli�èa, ki ga je shranilalgoritem, opisan v koraku 5.7 � Risanje numerièno dobljene KMD. Koraki 1 do6 se ponavljajo za vseh sedem geometrijskih oblikpreizku�anca, s èimer ovrednotimo razlièna podroèjav diagramu deformacijskih stanj.

Med analiziranjem nastanka lokalizacije oz.zloma na posameznem preizku�ancu po zgoraj opisanimetodi se lahko pri razliènih vozli�èih v enakemtrenutku pojavi izrazita sprememba sj&& . Pri nadaljnjianalizi ugotovimo, da so to sosednja vozli�èa, ki senahajajo v podroèju nastanka lokalizacije oz.poru�itve preizku�anca. Zato je na sliki 9 pri doloèenigeometrijski obliki vzorca prikazanih veè kritiènihtoèk.

3 PRIMERJAVA PREIZKUSNO IN NUMERIÈNODOBLJENE KRIVULJE MEJNIH DEFORMACIJ

Za doloèanje numeriène krivulje mejnihdeformacij, smo uporabili enak material preizku�ancakakor pri preizkusnem doloèanju KMD. Preizkusnidel je bil izveden na orodju za Marciniakov preizkus,opremljen z optiènim sistemom za vrednotenjepodatkov v Laboratoriju za preoblikovanje naFakulteti za strojni�tvo v Ljubljani.

Slika 9 prikazuje preizkusno in numeriènodobljene toèke KMD za vroèo cinkano jeklenoploèevino. Spodnja èrta na diagramu predstavljalokalizacijo, zgornja pa poru�itev preizku�anca pripreizkusnem doloèanju KMD.

Z opisanim numeriènim kriterijem zadoloèevanje KMD napovedujemo zaèeteklokalnega stisnjenja na preizku�ancu. Pri analizidiagrama ugotovimo, da se leva stran numeriènodobljenih toèk, v primerjavi s preizkusom, nagiba knapovedovanju pretrga preizku�anca. Izomenjenega lahko ugotovimo, da z numeriènimkriterijem ne moremo natanèno doloèiti zaèetkalokalizacije na preizku�ancu pri vseh geometrijskihoblikah vzorcev. Problem bi re�ili s pogostej�imshranjevanjem tanj�anja posameznih vozli�è napreizku�ancu, predvsem v trenutku nastankalokalizacije.

written that compare the origin times of maximal valueof the second derivative for particular specimennodes and then write down the origin time and thenode number at which the maximal value of thesecond derivative first appeared (at minimum time).6 � Recording the main strains in the transversal(j

1) and the longitudinal (j

2) direction of the node.

The main strains are written down in transversal andlongitudinal directions for the node, and saved withthe algorithm described in step 5.7 � Plotting the numerical FLC. Steps 1 to 6 are repeatedfor each of the seven specimens� dimensions in order toevaluate the different areas in the FLD diagram.

During the analysis the onset of neckingand fracture in the particular specimens by the meth-odology described above can appear at differentnodes at the same moment expressive variation ofthe sj&& . However, with further analysis this node wasdetermined as adjacent nodes, that are in the area,where the necking or failure on the specimen origi-nates. Therefore, more critical points for the definedspecimen geometry are shown in Figure 9.

3 COMPARISON OF THE EXPERIMENTAL ANDNUMERICALLY OBTAINED FLC

The numerical and experimentaldeterminations of the FLC were performed with thesame material properties. The experiments werecarried out on the Marciniak test tool and opticaldata-acquisition equipment in the FormingLaboratory, Faculty of Mechanical Engineering,Ljubljana.

The experimental and numerically obtainedpoints of the FLC for hot-galvanized steel are shown inthe Figure 9. The lower line on the diagram representsthe localization, whereas the upper line representsspecimen failure by the experimentally defined FLC.

With this numerical approach for thedetermination of the FLC the onset of necking onthe specimen is predicted. Furthermore, it is evidentfrom the diagram that the left side of the numericallyobtained points tends to predict specimen fracturein comparison with the experimental test. However,it could be established that it is not possible toaccurately determine the onset of necking on thespecimen for all the analysed geometries using thenumerical criterion. The problem could be solvedwith a higher frequency of data saving for all thenodes on the specimen, especially at the momentwhen the necking occurs.



Numerièno dobljene toèke mejnih deformacij sepojavljajo med podroèjem, ki ga zajemata KMDlokalizacije in zloma preizku�ancev pri preizkusu. Kerse podroèje numeriène krivulje nahaja v kritiènempodroèju preizkusne krivulje, jo lahko vsekakoruporabimo za oceno preoblikovalnosti materiala.

Do odstopanj med numerièno in preizkusnodobljenimi toèkami mejnih deformacij lahko pridezaradi vizualnega pregleda posnetkov zaèetkalokalizacije in kasneje zloma na preizku�ancu pripreizkusnem doloèanju toèk, kar ima lahkopristransko naravo. Z uporabo pogostej�egashranjevanja posnetkov bi lahko omenjeni pristranskivpliv zanemarili.

Pri doloèevanju KMD z numeriènosimulacijo pa se izognemo pristranskemu vplivudoloèitve lokalizacije na preizku�ancu. V tem primeruje natanènost numerièno analizirane lokalizacije napreizku�ancu odvisna od gostote shranjevanjapodatkov med simulacijskim postopkom.

Primerjava med preizkusno in numeriènodobljeno KMD je bila izvedena na podlagi trehrazliènih oblik vzorcev (na diagramu slika 9: levo -platina �irine 90 mm, sredina � platina �irine 130 mmin desno � rondela). Zaradi natanènej�ega potekanumeriène KMD so v diagram (sl. 9) vnesene tudimejne deformacije preostalih numerièno analiziranihpreizku�ancev (platina �irine: 95 mm, 135 mm, 150

Numerically obtained points of the strain limits appearbetween the area that is limited with necking and thefracture limit curve by an experimental determinationof the FLC. Since the numerical curve is placed in acritical area of the experimental curve it can be anywayused for the estimation of the material formability.

The deviation between the numerically andexperimentally obtained strain limit points can berelated to a visual data evaluation of the onset ofnecking and the later fracture in the specimen by anexperimental determination of the points, which canhave a subjective nature. The above-mentionedsubjective influence can be neglected by usingrepeated data saving.

The subjective influence of the neckingdefinition on the specimen is negligible for anumerical determination of the FLC. In this case theaccuracy of the numerically analysed necking onthe specimens depends on the frequency of thesaved data.

The comparison between the experimen-tally and numerically obtained FLC is based on threedifferent test geometries (Figure 9: left � specimenwidth of 90 mm, middle � specimen width of 130 mmand right � blank of 200 mm in diameter). The limitstrains of the other numerically analysed specimenswith widths of 95 mm, 135 mm, 150 mm and 165 mmare also entered in the diagram (Figure 9) for an ac-

Sl. 9. Primerjava numerièno in preizkusno dobljene KMDFig. 9. Comparison of numerical and experimental obtained FLD

0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

-0,4 -0,2 0 0,2 0,4

eksperimentalni rezultati - za�etek lokalizacije / experimental results - onset of neckingeksperimentalni rezultati - poru�itev / experimental results - fracturenumeri�ni rezultati / numerical results

�1

�2

poru�itev / fracture

lokalizacija / neckingj

1

j2

èe

èn



curate definition of the numerical FLC. A large numberof specimens with widths around 130 mm would beneeded to analyse the accurate position of the FLC,because a small difference in the specimen geom-etry can have an enormous influence on changes tothe strain path against time.

4 CONCLUSION

The most important advantage of thepresented numerical method is its potential forautomation and the evaluation of a large number ofdifferent specimen geometries. Therefore, the methodis able to quickly and reliably predict the forminglimit curve.

In spite of the differences between thenumerically and experimentally obtained points thefinal results are acceptable, because the numericallyobtained FLC is in good agreement with theexperimentally obtained curve over the whole rangeof the forming limit diagram.

Satisfactory results are obtained with lesscost and in a shorter time than with experiment. Ofcourse, the more accurate are the material data forthe input of the numerical program, the better is theobtained forming limit curve.

In future research work we will determine, withthe above-mentioned numerical approach, the FLCsfor other materials with diverse mechanical properties(aluminium, titanium, stainless steel, etc.). In this waythe general validity of the presented methodology willbe confirmed. In the case of deviations the reasons forthese anomalies will be analysed as well.

mm in 165 mm). Za natanènej�o doloèitev lege KMDbi bilo potrebno analizirati veèje �tevilopreizku�ancev s �irinami okoli 130 mm, saj �e takomajhna sprememba geometrijske oblike vzorcavpliva na spremembo poteka deformacije vodvisnosti od èasa.

4 SKLEP

Najveèja prednost predstavljene metode jemo�nost avtomatizacije in vrednotenje velikega�tevila razliènih geometrijskih oblik preizku�ancev.Zato je metoda zmo�na hitro in zanesljivo napovedatikrivuljo mejnih deformacij.

Kljub temu, da prihaja do razlik mednumerièno in preizkusno dobljenimi toèkami,lahko rezultate �tejemo kot dobre, saj senumerièno dobljena KMD dobro prilegapreizkusni KMD na celotnem obmoèju diagramamejnih deformacij.

Dobljeni so zadovoljivi rezultati z ni�jimistro�ki in v kraj�em èasu kakor pri preizkusu. Znatanènej�im vnosom materialnih podatkov vnumerièni program bi dosegli popolnej�o krivuljomejnih deformacij.

V nadaljevanju raziskovalnega delabomo z opisano metodo doloèili �e KMD zamateriale, ki imajo drugaène mehanske lastnosti(aluminij, titan, nerjaveèe jeklo itn.). S tem bomodobili potrditev o splo�ni veljavnosti takodobljenih krivulj mejnih deformacij, v primeruodstopanj pa bomo sku�ali ugotoviti vzroke zanjihov nastanek.

5 LITERATURA5 REFERENCES

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[2] S. P. Keeler (1964) Plastic instability and fracture in sheet streched over rigid punches. ASM Trans. 56, 25-48.[3] G. M. Goodwin (1968) Application of strain analysis to sheet metal forming in the press shop. SAE paper

No. 680093[4] R. Hill (1952) On discontinuous plastic states, with special reference to localized necking in thin sheets,

J. Mech. Phys. Solids 1, 1952[5] H. W. Swift (1952) J. Mech. Phys. Solids, 1 (1952) 19.[6] Z. Marciniak, K. Kuckzinsky (1967) Limit strains in the processes of stretch-forming sheet metal, Int. J.

Mech. Sci. 9 (1967) 609-620[7] F. Cayssials (1998) A new method for predicting FLC, Proc. Of 20th IDDRG, Genval, Belgium 1998, p. 443-

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sistemi in proizvodna kibernetika, 3. del, Identifikacija preoblikovalnosti tanke ploèevine, Ljubljana.



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[10] Z. Marciniak, K. Kuczinski, T. Pokora (1973) Influence of the plastic properties of the material on theforming limit diagram for sheet metal tension, Int. J. Mech. Sci., 15 (1973), p. 789-805.

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[14] J. Kadivnik (1982) Uporaba grafometriène metode pri komparativni analizi preoblikovanja tanke ploèevineza izdelavo velikih karoserijskih delov, magisterij, Fakulteta za strojni�tvo, Ljubljana.

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[16] F. Gologranc, K. Kuzman (1989) Tehnika preoblikovanja � stanje in smeri razvoja. V: Obdelovalna tehnika:1. seminar, Fakulteta za strojni�tvo, Ljubljana.

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[19] F. Barlat (1989) Forming limit diagrams-prediction based on some microstructual aspects of materials, TheMinerals, Metals, Materials Society.

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Naslov avtorjev: Ale� Petekdr. Toma� Pepelnjakprof.dr. Karl KuzmanUniverza v LjubljaniFakulteta za strojni�tvoA�kerèeva 61000 [email protected]@[email protected]

Authors� Address: Ale� PetekDr. Toma� PepelnjakProf.Dr. Karl KuzmanUniversity of LjubljanaFaculty of Mechanical Eng.A�kerèeva 6SI-1000 Ljubljana, [email protected]@[email protected]

Prejeto: 18.2.2005

Sprejeto: 25.5.2005

Odprto za diskusijo: 1 letoReceived: Accepted: Open for discussion: 1 year

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Nova metoda za doloŁevanje krivulje mejnih deformacij v