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A methodology for evaluating sheet formability combining the tensile test with the M–K model

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Materials Science and Engineering A
528 (2010) 480–485
Contents lists available at ScienceDirect
MaterialsScienceandEngineeringA
journal homepage: www.elsevier.com/locate/msea
A methodology for evaluating sheet formability combining the tensile test withthe M–K model
Cunsheng Zhang
a
,
∗
, Lionel Leotoing
b
, Guoqun Zhao
a
, Dominique Guines
b
, Eric Ragneau
b
a
Shandong University, Key Laboratory for Liquid-Solid Structural Evolution and Processing of Materials (Ministry of Education), Jinan 250061, Shandong Province, PR China
b
Université Européenne de Bretagne, INSA-LGCGM - EA 3913, 20, avenue des Buttes de Coësmes 35043, Rennes Cédex, France
a r t i c l e i n f o
Article history:
Received 7 June 2010Accepted 2 September 2010
Keywords:
Forming limit curves (FLCs)Marciniak and Kuczynski (M–K) modelMarciniak testDigital image correlation (DIC)
a b s t r a c t
This paper proposed an approach for evaluating the sheet formability by combining the tensile test withtheﬁniteelementMarciniakandKuczynski(M–K)model.Firstly,thetensiletestwithanotchedspecimenwascarriedouttoidentifyanappropriateconstitutivelawforanAA5086sheet.AmodiﬁedLudwick’slawwasusedtodescribeitsformingbehavior.Atechniqueofdigitalimagecorrelationassociatedwithahigh-speed camera was applied to evaluate specimen’s surface strains during the experiments. The inverseanalysis was performed to identify the parameter values in the constitutive law. The initial geometricalimperfection factor in the M–K model was determined. Then by using the commercial ﬁnite elementsoftware ABAQUS, the M–K model was simulated to evaluate numerically the sheet formability of thealloy. By means of a user-deﬁned FORTRAN subroutine UHARD, the constitutive law was implanted intoABAQUS.Differentstrainstateswereobtainedbychangingdisplacementratiosandforminglimitcurves(FLCs) of the sheet were determined too. Finally, an experimental procedure based on the modiﬁedMarciniak test was carried out and the FLCs were obtained experimentally. The comparison betweenthe numerical and experimental results showed that the approach developed in this paper could give anappropriate prediction of FLCs.
© 2010 Elsevier B.V. All rights reserved.
1. Introduction
Forsheetmetalforming,forminglimitcurves(FLCs)areanefﬁ-cient diagnostic tool for evaluating sheet formability and manymethods have been developed to determine the FLCs [1–3]. How-
ever, the determination of FLCs is a complex task, there is nowell-established experimental or numerical procedure for itsdetermination [4].ForexperimentalpredictionsofFLCs,twomainkindsofformingtests have been developed, the so-called out-of-plane stretching(e.g. the Nakazima test [5], the Hecker test [6]) and the in-plane
stretching(e.g.theMarciniaktest[7]).Duringout-of-planestretch-
ing, as illustrated in Fig. 1(a), the blank is deformed under triaxial
stresswhileduringin-planestretching,thestressperpendiculartothe sheet surface is small compared to the stresses in the planeand could be neglected, hence the sheet is under near plane stressconditions in the central part (see Fig. 1(b)).
ExperimentalmethodisabasicwaytoobtainFLCsofsheetmet-als. However, there is no precise standard to detect the onset of localizedneckingandconstructmorestableandreproductibleFLCs,inadditiontoISO12004whichisgenerallyconsideredastoofuzzy[8]. Moreover, it is a very time consuming procedure to establish
∗
Corresponding author. Tel.: +86 53181696577; fax: +86 53188392811.
E-mail address:
zhangcs@sdu.edu.cn (C. Zhang).
FLCsandthescatterinexperimentaldataforagivensheetisusuallylarge [9]. Especially, little research on sheet formability has been
reported at high strain rates due to the difﬁculty in carrying outthe experiments. As a result, signiﬁcant efforts have been made ondeveloping more analytical or numerical models for constructionof FLCs.From the viewpoint of numerical research, due to the devel-opments in the methods of modeling and simulation as well asin computational facilities, numerical predictions of FLCs havebecome more attractive, and FE method has been selected to sim-ulate the necking process.Using the LDH (Limiting Dome Height) test, Narasimhan [10]haspredictedtheonsetoflocalizedneckingbythethicknessstraingradient across neighboring regions. When the thickness gradientin adjoining regions was 0.92, localized necking was assumed tooccur. The predicted and experimental FLCs of a steel sheet are ina good agreement.Based on the Marciniak test, Petek et al. [1] put forward a new
method by evaluating the thickness strain as a function of time aswell as the ﬁrst and second time derivative of the thickness strain.They proposed that the maximum of the second temporal deriva-tive of the thickness strain corresponds to the onset of localizednecking.Volk [11] identiﬁed the onset of localized necking by experi-
mental and numerical methods. With calculated strain rates, theidentiﬁcation was carried out with the two following main effects:
0921-5093/$ – see front matter
© 2010 Elsevier B.V. All rights reserved.
doi:10.1016/j.msea.2010.09.001
C. Zhang et al. / Materials Science and Engineering A
528 (2010) 480–485
481
Fig. 1.
Schematic layouts of out-of-plane and in-plane stretching.
increase of points number with high strain rate (in the localiza-tion region) and decrease of the strain rate outside the localizationbands.In addition, the Marciniak and Kuczynski model (known as theM–K model) is a widely used analytical one which can help toreduce the experimental effort of formability characterization aswell as to predict FLCs of sheet metals [12]. So far, the M–K model
has also undergone great improvement. However, for a complexconstitutive law, the analytical M–K model does not work wellbecausetheinherentsystemequationscannotbeeasilyresolved.Incontrast,withnumericalmethods,abovelimitscouldbeovercomeby implanting any complex constitutive law into FE code [13].Therefore, the numerical M–K model is also simulated to con-struct the FLCs of sheet metals. Banabic et al. [14] determined FLCs
by simulated the M–K geometrical model with an inclined groove.For the left-hand side of FLCs, the imperfection orientation wastakenaccordingtoHill’szeroextensionassumption.Agoodcorrela-tionbetweenpredictedandexperimentalresultshasbeenobtainedfor right-side hand of FLCs, while in left-hand area the predictedFLCs underestimated the experimental ones. Recently, Zhang [13]simulated the M–K model with ABAQUS by means of the imple-mentationofdifferenthardeninglaws(Swift’slaw,Johnson–Cook’slaw and Ludwick’s law), it is found that hardening laws inﬂuencegreatly the determination of FLCs.However, the initial geometrical imperfection factor
f
0
in theM–K model is an uncertain one. Generally, its value is determinedby making the best ﬁt between the numerical and experimen-tal results. Moreover, an appropriate constitutive law is a key toobtaining the practical prediction of FLCs. Hence, the paper beginswith the tensile test for an AA5086 sheet. An inverse analysis isapplied to identify ﬂow behaviors of this aluminum sheet and thecorresponding parameters in the constitutive law are determined.Furthermore, the initial imperfection factor
f
0
in the M–K modelis determined for this given sheet, and the model is simulated toconstruct the FLCs of this sheet. Finally, an apparatus based on theMarciniak test is developed to experimentally construct the FLCs.Aquasi-staticexperimentalprocedureiscarriedouttovalidatetheproposed numerical approach.
2. Tensile test
In this part, the tensile test is carried out on a computer-controlled servo-hydraulic testing machine. A technique of digitalimage correlation (DIC) associated with a high-speed camera isapplied to evaluate surface strains and a complete procedure isbuilt to detect the onset of localized necking during the experi-ments. Then the use of inverse analysis is performed to identify aconstitutive law for this aluminum alloy sheet.
Fig. 2.
Geometry and dimension of a notched specimen (length in mm).
2.1. Experimental tensile tests
Forthetensiletestinthiswork,aspecimenisspeciallydesignedwith a notch which may result in fast necking initiation and facil-itate the registration of a series of consecutive images of thelocalized region. The geometry and dimension of the specimensis shown in Fig. 2. The thickness of this sheet is 2.0mm.
To capture the consecutive images during the experiments,a Fastcam ultima APX-RS digital CMOS camera associated witha macro lens is used. The commercial digital imaging programCORRELA2006, developed by LMS at the University of Poitiers, isemployed to perform correlation analysis in this work. The DICprogram produces the information of the surface strains on thespecimen. To ﬁnd a representative procedure for analyzing thenecking progress, the time-sequence of equivalent plastic strainproﬁlesalongthelongitudinalaxisofspecimenatdifferentinstantsof time is displayed in Fig. 3.From the ﬁgure, it is clearly observed that at early stage of theforming process, a quasi-homogeneous deformation is distributedalongthelongitudinalaxisofspecimen.Thenthesubsequentdevel-opment of deformation is concentrated to the central part of thespecimen notch, and plastic strain is localized to a smaller andsmaller region. When the strain increment in the localized zoneexceeds by 7 times that in non-localized zone, a critical momentcould be determined corresponding to the onset of localized neck-ingandtheprincipalstrains(
−
0.067,0.30)inthelocalizedzoneareretained as the limit strains to form one point of FLCs of this sheet.A strain path of
−
0.223 for this tensile test could be calculated bythe ratio of minor strain and major strain. Because of its irregularspecimen geometry in the present tensile test, the strain path isclearly different from that of a general uniaxial tension, which isabout
−
0.5.
Fig. 3.
Strain proﬁles along the longitudinal axis of specimen.
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C. Zhang et al. / Materials Science and Engineering A
528 (2010) 480–485
Fig. 4.
Experimental and identiﬁed curve of force versus displacement.
2.2. Identiﬁcation of constitutive model
By use of the tensile test with a notched specimen, it is difﬁcultto identify the material’s ﬂow behavior with conventional ana-lytical methods due to the irregular specimen cross-section [15].Nowadays, the inverse methods are intensively used to adjust thematerialparametersformoreandmorecomplexconstitutivelawsor irregular specimen geometry. The basic concept of an inverseanalysisforparameteridentiﬁcationistoﬁndoutasetofunknownmaterial parameters in constitutive equation thanks to a FE simu-lation of the test.To describe its elasto-plastic behavior for this given sheet, ageneral constitutive model¯
=
0
+
K
¯
ε
n
(1)isproposedtobeidentiﬁed.Here,
K
and
n
arematerialparameters.The detail identiﬁcation procedure can be found in Diot’s paper[16]. For the uniaxial tensile test under 10mm/s at 20
◦
C, the iden-tiﬁed results with the inverse analysis are:
0
=147MPa,
K
=870MPa,
n
=0.335. The comparison between experimental and calcu-latedloadsversusdisplacementisillustratedinFig.4.Itcanbeseen
that there is a good agreement between the experimental curvesandtheoneidentiﬁedbyinverseanalysis.Tocomparewithexper-iments,theidentiﬁeddatawillbeusedforthefollowingnumericalprocedure.
3. M–K model
In this part, the M–K model is simulated with the commercialﬁnite element code ABAQUS to numerically construct the FLCs of thestudiedsheet.Withexperimentalresultsobtainedbythetensiletest,theinitialgeometricalimperfectionfactor
f
0
intheM–Kmodelisdetermined,andtheconstitutivelawidentiﬁedbyaboveinverseanalysis is implanted into ABAQUS with its subroutine UHARD.
3.1. Construction of the M–K model with ABAQUS
Similarly as the analytical M–K model, an initial defect in thesheet is characterized by two different zone thicknesses in the FEmodel. Here, it is assumed that the imperfection zone (zone b) isperpendicular to the principal axis-1 [17]. Due to symmetry, only
one half of the entire model in the thickness is considered for thisFE analysis, as shown in Fig. 5(a).
Thesheetismeshedbyhexahedralelements.Tocomparedefor-mation states in the two different zones, two different referenceelements are required. One of the elements is placed in zone a(Element A), while the other is in zone b (Element B). Essentialboundary conditions are imposed by displacement constraints oncertain surfaces of the model (see Fig. 5(b)).
The elasticity of this material is deﬁned with the Young’s mod-ulus of 70500 MPa and the Poisson’s ratio of 0.33. By means of auser-deﬁned subroutine UHARD, the constitutive model identiﬁedby above inverse analysis is implanted into ABAQUS.Because of the relatively smaller thickness in zone b, the equiv-alent plastic strain in zone b is greater than that in zone a. Themaximum values occur at the center of the model, while fartherfromthecenter,thestrainreducesgradually.Fig.6(a)clearlyshows
theevolutionsoftheequivalentplasticstrainofElementAandEle-ment B. As observed from this ﬁgure, the strain histories from thetwo elements are relatively similar until they diverge at approxi-mately
t
=8s. At this stage, the equivalent plastic strain in ElementB rises rapidly while that in Element A shows a relative saturation.When the equivalent plastic strain increment in Element Bexceeds by 7 times that in Element A (corresponding to
t
0
inFig.6(b)),localizedneckingisassumedtooccurandtheﬁnalmajorand minor strains of Element B calculated by linear interpolationare noted as the limit strains for construction of FLCs.
3.2. Identiﬁcation of the initial imperfection factor f
0
ShallowinitialgroovesaresufﬁcienttocauselocalizationintheM–K model [18]. Generally, the value of the initial imperfection
factor
f
0
is chosen to make the best ﬁt between the numerical andthe experimental results, and to some extent, this value denotesthe level of sheet formability. Hence, to characterize the practicalformability of a given sheet, an appropriate value of
f
0
should beidentiﬁed.Different strain states could be covered by imposing differ-ent ratios of displacements in the 1 and 2 directions as shown in
Fig. 5.
FE M–K model and corresponding boundary conditions in ABAQUS. (a) FE model in ABAQUS. (b) Boundary condition in the M–K model.
C. Zhang et al. / Materials Science and Engineering A
528 (2010) 480–485
483
Fig.6.
Evolutionsoftheequivalentplasticstrainanditsincrementratiooftwoelementsselected.(a)Evolutionoftheequivalentplasticstrainoftwoelements.(b)Evolutionof the equivalent plastic strain increment ratio.
Fig. 7.
FLCs obtained with the FE M–K model with different imperfection factors.
Fig. 5(b). Here, the displacement
u
in direction 1 is ﬁxed at 60mm;byvaryingthedisplacement
v
indirection2,strainstatechanges.Tochooseanappropriateimperfectionfactor,theM–Kmodelwithdif-ferent imperfection factor
f
0
(0.92, 0.95, 0.98) is simulated and theFLCsareobtainedasshowninFig.7.Regularshapesofforminglimit
curvearefoundfromthisﬁgure.Incomparisonwith
f
0
=0.92,thereareincreasesof45%,15%,and38%inmajorstrainfor
f
0
=0.98underuniaxial tension, plane strain and equi-biaxial stretching condi-tions, respectively. Moreover, the level of critical strains for theM–K model with
f
0
of 0.92 approaches to that of the experimentaltensiletest(
−
0.067,0.30).Therefore,forthissheetformability,theinitial imperfection factor with the value of 0.92 is suitable.
4. Experimental validation
The necessity to verify the analytical and numerical predictionsleads to a further study of the precision and efﬁciency of deﬁnitionofFLCswithexperiments[1].Duringtheexperiments,oneormore
high-speed cameras are used to view the sample surface and takeconsecutive images during experiments. In contrast to the Nakaz-ima test, the investigated region in the Marciniak test remains ﬂatduringtheexperiment,asillustratedinFig.1.Becausethiscaseisa
2D-application,strainscanbemeasuredwithonlyonecamera.Thisisanimportantadvantageofthismethodandanimportantreasonfor us to choose the Marciniak test to determine FLCs in this work.Aquasi-staticexperimentalprocedureiscarriedouttoexperimen-tally construct FLCs of the AA5086 sheet and compare with abovenumerical results.
4.1. Experimental preparations
In this work, a new experimental apparatus based on theMarciniak test is developed, which includes a reverse experimen-tal setup and an image acquisition system, as shown in Fig. 8(a). In
this reverse setup, a bell jar is designed to connect the die withthe crosshead of tensile machine. The purpose of this design isto prevent or minimize the vibrations and ensure the rigidity of the experimental setup throughout static and dynamic tests. Onthe bell jar, there are two small windows, one is for installing theoptic mirror and another permits that the reﬂected light from thespecimen surface goes through then focuses on the mirror. Duringtheexperiments,thediewiththeclampedspecimenmovesdown-wards and the ﬁxed punch stretches the sheet. With this reverse
Fig. 8.
Reverse experimental setup based on the Marciniak test. (a) 3D model of experimental setup. (b) Experimental apparatus.
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C. Zhang et al. / Materials Science and Engineering A
528 (2010) 480–485
Table 1
Specimen dimensions used for the experiments.
W
(mm) 10 20 30 40 45 48 50 52 55 58 60 80 100
R
(mm) 50
Rc
(mm) 70
Rm
(mm) 26.5
Re
(mm) 10
experimental setup, in which the specimen is formed over thepunchuntilfractureappearsonthespecimensurface,thedistancebetween the mirror and the specimen remains nearly constantthroughout the test. This allows the camera to be focused on thespecimen surface before the test and take sequential pictures.In order to cover different strain states, ranging from uniaxialthrough plane strain to equi-biaxial stretching, different specimengeometries are used. In this work, all test samples are shaped bycutting strips of different widths
W
in a circular ﬂange (Table 1),according to Fig. 9(a). To assure the occurrence of the maximal
strains (to trigger localization) on the central part of the blank, thespecimens are designed with a reduced central thickness (0.8mm)compared to the thickness of the sheet (1.5mm) and the clampingpart with a thickness of 2.0mm.Before the test, the experimental specimen is painted with arandom speckle pattern for DIC analysis. The surface which hasan applied speckle pattern should be away from the punch con-tact surface. Finally, the painted specimen is clamped betweenthe blankholder and the die. The bell jar, together with thedie, moves down at a crosshead traveling speed of 500mm/min.Here, a quasi-static procedure at 20
◦
C is carried out to test theexperimental setup and compare with above numerical proce-dure.Thecomparisonbetweenanundeformedandcrackspecimenin the experiments is shown in Fig. 10. With the DIC programCORRELA2006, a correlation analysis is carried out on the crackspecimen and surface strains are calculated.
4.2. Experimental results
As an illustrated example, the specimen of Fig. 10(b) is chosen.
Inordertocarryoutastudyabouttheplasticinstabilityassociatedwith the onset and progress of necking in the sheet, two points arechosen. Point B is in (or near) the zone where the rupture occurswhile Point A is outside of the zone, as shown in Fig. 11.TheevolutionofthemajorstrainsofPointAandPointBisplot-ted in Fig. 12. A similar evolution in these two curves is observed
up to a moment of about
t
=0.9s, after which they diverge rapidly.Here, the criterion widely used in the M–K model is chosen to pre-
Fig. 9.
Specimens specially designed in the experiments. (a) Specimen geometrydesigned for the experiments. (b) Blank with non-uniform thickness in the experi-ments.
Fig. 10.
Specimens used in the experiments. (a) Undeformed specimen. (b)Deformed specimen.
Fig. 11.
Deformed specimen with rupture.
dict the onset of localized necking. When the equivalent plasticstrain increment ratio between Point A and Point B attains 7, theonsetoflocalizedneckingisassumedtooccurandthecorrespond-ing major and minor strains (
−
0.0036, 0.20) of Point B are retainedas a point on the FLC.To provide sufﬁcient and reliable data, for each specimen, atleast two tests are performed under identical conditions.
Fig. 12.
Evolution of major strains for the two Point A and Point B.

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