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curve (FLC) characterization of metallic sheets
Ibrahim Zidane, Dominique Guines, Lionel Leotoing, Eric Ragneau
To cite this version:
Ibrahim Zidane, Dominique Guines, Lionel Leotoing, Eric Ragneau. Development of an in-plane biaxial test for forming limit curve (FLC) characterization of metallic sheets. Measurement Science and Technology, IOP Publishing, 2010, 21, pp.1-11. �10.1088/0957-0233/21/5/055701�. �hal-00981719�
haraterization of metalli sheets
I. Zidane, D. Guines, L. Leotoing, E. Ragneau
UniversiteEuropeennedeBretagne,Frane,INSA-LGCGM-EA3913,
20,avenuedesButtes deCoesmes35043RENNES Cedex
E-mail: dominique.guinesinsa-rennes.fr
Abstrat. Themainobjetiveofthisworkisto proposeanewexperimentaldevie
able to give for a single speimen a good predition of rheologial parameters and
formabilityunderstatianddynamionditions(forintermediatestrainrates). Inthis
paper,wefousontheharaterizationofsheetmetalforming. Theproposeddevieis
aservo-hydraulitestingmahine provided withfour independentdynamiatuators
allowing biaxial tensile tests on ruiform speimens. The formability is evaluated
thanks to the lassial forming limit diagram (FLD) and one of the diÆulties of
thisstudywasthedesignofadediatedspeimenforwhihthenekingphenomenon
appearsin its entral zone. Ifneking isloatedin theentralzone ofthe speimen,
then thespeed ratiobetween the two axes ontrols thestrain path in this zone and
a whole forming limit urve an be overed. Suh a speimen is proposed through
a numerial and experimental validation proedure. A rigorous proedure for the
detetionofnumerialandexperimentalformingstrainsisalsopresented. Finally,an
experimentalforminglimiturvedisdeterminedandvalidatedforanaluminiumalloy
dediatedtothesheetformingproesses(AA5086).
Keywords: biaxial tensile test, ruiform speimens, forming limit diagram (FLD),
dynami tests, DigitalImage Correlation (DIC)
Submittedto: Meas. Si. Tehnol.
1. Introdution
Sheet metal forming is one of the most ommon metal proessing operation. More
and more preision is required relatively to both the geometry and the mehanial
propertiesofthenalprodut. Forthisreason,the traditionaltrial-and-errormethodof
optimizingsuh metalforming operations isapparently ineÆient. Also,the modelling
of material behavior is a very eÆient way of reduing the time and osts involved in
optimizingmanufaturing proesses. Simulation of sheet metal forming, mainly based
on nite element (FE) modelling, requires to know preisely the onstitutive law and
at strain, strain rate and temperature ranges orresponding to forming proesses. As
most metal forming operations are arried out under biaxial states of stress, limiting
the evaluationof material'smehanialharateristis touniaxial tensile tests an lead
toamisrepresentationofthe behaviorofthe material. The useof morerealistiloading
during the test suh as the introdution of biaxial loading onditions leads to a more
auraterepresentationoftheexpetedbehaviorofthematerialbyonsideringpreisely
anisotropy eets, for example.
The general purpose of this work is to propose an experimental test and the
assoiatedspeimen shape fordeterminingthe wholeelasto-plasti behaviorof metalli
sheets inluding hardening, anisotropi behavior and formability limits. As explained
above, to be in aordane with operating onditions enountered in metal forming
proesses, theses haraterizations must be performed for biaxial loadings and for
large ranges of strain rate and temperature. At the present time, the identiation
of hardening parameters under mono-axial, stati or dynami loadings are lassially
performed by means of numerial modelling used with optimization methods in the
so-alled inverse analysis proedure ([1℄, [2℄). This paper fouses on the development
of both an experimental test and the assoiated speimen shape appropriate for the
determinationof a wholeforming limiturve for stati loadings.
Awell-known methodofdesribingtheformabilityof materialsistheforminglimit
diagram(FLD).In FLDs,aFLC(FormingLimitCurve)representsaplot ofmajorand
minor available prinipal strains in the plane of the deformed sheet orresponding to
the ourrene of the neking. The FLD determinationis usually obtained from stati
onventional tests(Mariniaktest[3℄, Nakazimatest [4℄orbulgetest[5℄). Insuhtests,
only linear strain paths (ratio between major and minor strains) an be realized. The
paths are dened from the speimen or tool geometry and many dierent speimens
are required to over the whole forming limiturve. Dependingon the onsidered test,
the fritioneet between the speimenand the toolaninuene the preditionof the
material formability and it is still a diÆulty to evaluate the oeÆient frition value.
Moreover, strain-ratesensitivityhas been identiedasanimportantfatordetermining
formabilityof sheet metaland an alter substantially the level and the shape of FLCs.
A review of urrently availableliterature shows that relatively littleattention has been
paid to models taking strain rate into aount. Analytially,the strain rate sensitivity
hasbeenstudiedmainlythroughthelassialM-Kmodel[6℄. Intheseanalytialmodels,
the strain hardening behavior and the strain rate sensitivity of materials are generally
represented by simplistipower laws. Butit has been shown that these simplistilaws
are not well adapted, espeially for aluminium alloys, and only permit the study of
the inuene of the strain rate sensitivity index but not the inuene of the value of
the strain rate and temperature [7℄. Experimentally, it is relatively diÆult to adapt
the onventional tests to dynami loadings due to the dynami response of the whole
testingdevies. Veryfewexperimentaldata areavailableand noorrelationshavebeen
presented between experimental results and those of numerial oranalytial preditive
In order to overome the drawbaks of the above mentioned tests, a ruiform
speimenould beaninteresting alternativeifthe strain pathatthe onsetofnekingis
diretlyimposedbytheontrolofthetestingmahine,independentlyfromthespeimen
shape. Moreover,forsuhspeimen,thefritionannotalterbothformabilityandstrain
path. Nevertheless,thiskindofspeimenhasneverbeenusedinformabilitystudieseven
if many authors intensively used it for other mehanial haraterizations (fatigue [8℄,
reep [9℄ oryieldriteriaand hardeninglaws [10℄). Although ruiformspeimens have
been investigated quite extensively, no standard geometry exists to this day [11℄ and
the design of the speimen shape is still the main diÆulty that restrits appliations
for the ruiform biaxialtensiletest.
The developed test mahinestoproduebiaxialloadingonruiformspeimenare
either stand-alone biaxial testing mahines or link mehanism attahments for biaxial
testing. ThedeviedesignedbyMakindeetal. [12℄onsistedoffourhydrauliatuators
with a apaity of 250kN. Two atuators were used on eah axis to ensure that the
enter of the speimen did not move during testing. The results from this researh
were used to develop an optimum speimen for low strains. Using the same testing
apparatus Green et al. [13℄ have investigated the elasto-plasti behaviour of an 1145
aluminiumsheetalloy. Flatruiformspeimenshavebeentestedunderin-planebiaxial
loadingonditions toobtain boththe plastiwork ontoursand the biaxialowurves.
A biaxial extensometer is used to measure and ontrol the strains in the entral area
of the speimen along eah loading diretion. A nite element analysis of eah test
was arried out. The numerial FE modeling ensure that the omputed fore in the
arms versus strain in the enter of the sample urves oinide with the orresponding
experimental ones. Thusthe parametersof the four dierentphenomenologial models
of anisotropi plastiity and biaxial owurves are determined. A further stand-alone
devie developed by Boehler et al. [14℄ onsisted of four double ating srew driven
pistons, mounted on an otogonal vertial frame. The test speed varies between 0:003
and 0:3mm=min. This mahine was used to test various ruiform speimen types
to develop an optimum speimen design. The optimized speimen is performed to
identifyanisotropielastibehaviour. Kuwabaraetal. [15℄study theelastiandplasti
deformationbehavior of old rolled low-arbonsteel underbiaxialtension. The biaxial
tester was a servo type and opposing hydrauli ylinders were onneted to ommon
hydrauli lines so that the same hydrauli pressure was applied to eah. A load ell
was used in eah diretion to alulate the stress in the speimen. The strain was
alulated using strain gauges plaed in the gauge setion of the speimen. The ow
stresses measured were in relatively good agreement with those predited by Hosford's
yieldriterion. NaumenkoandAtkins[16℄performedbiaxialtensiletests onpre-raked
high-strength low-hardening aluminiumspeimens. They demonstrated that the eet
of biaxiality loading on the rak extension resistane R, was unexpetedly strong.
Shimamoto et al. [17℄ have developed a hydrauli biaxial testing system to perform
bothdynamiandstatitests. Fromdynamitests,theyinvestigatethe rakextension
ostassoiatedwithbuildingstand-alonetestmahines,standardtensileorompressive
testers have been onverted tobiaxialtest setup (Hoferlinet al. [18℄, Mohr and Mulalo
[19℄, Ferron and Makinde [20℄). Nevertheless, link mehanism attahments for biaxial
testing mahines seem not very eÆient to study the forming limitsunder various and
omplexloadingpathswhihwouldrequiretohangethegeometryofthesystem(length
of links, ...) for eah path. Moreover, in suh experimentaldevies, both high stiness
and natural frequenies are expeted so these systems are not appropriate for biaxial
tests and partiularly dynami ones. Stand-alone testing mahines seem to be more
appropriated to realize biaxial tests and more partiularly to obtain dierent strain
paths or various stress states of biaxialtension by hangingthe proportion of loads or
displaements of the two axes. Moreover, to perform both stati and dynami tests
with the same devie, hydrauli tehnology seems more appropriated, by using pump
pressure or aumulator pressure for stati tests and instantaneously releasing a great
quantity of high-pressure owaumulatedfor dynami tests.
In this work, a servo-hydrauli testing mahine provided with four independent
dynami atuators has been used. The testing devie is in a horizontal onguration
whih failitates aess for an easy setting of the speimen and allows a good follow-
up of the entral zone throughout the test by video reording. The enter point of
the speimen isalways maintainedstationarythroughout the test by aneÆient servo-
hydrauli ontrol. In order to determine the forming limits under omplex loading
paths,aspeimenshapehas tobedened. The maindiÆultytodesignanappropriate
ruiformspeimenshapeonsistsinforingtheonsetofnekingintheentralzoneand
not on the arms of the speimen. Moreover, for dynami tests, the maximum stiness
of thespeimen, theinitialone,has tobeadjustedaording tothe experimentalsetup
apaities in order to ontrol tensile test veloities. For the onsidered experimental
devie, the stiness of the speimens should not exeed 45kN=mm for dynami tests.
Others onsiderationslikemanufaturingonstraints must alsobetaken into aount.
In this study, dierent speimen shapes used in previous works are numerially
investigated through FE simulations. A modied ruiform speimen shape is then
proposedandthe best set ofgeometrialdimensionsisdenedfromaparametristudy.
Theproposed speimenshapeisthen numeriallyvalidated. Finally,the harateristis
of the developed biaxial system are presented and experiments are onduted on the
proposed ruiform speimen shape. The eÆieny of the proposed speimen shape is
alsovalidatedby experiments. The entralarea of the ruiformspeimen is lmed by
a amera and strain elds are determined by means of animage orrelation tehnique.
For dierent strain paths, experimental results, i.e., the strain level at the onset of
nekingfromtheruiformspeimenare omparedwith thoseof themore onventional
Mariniak test.
2. Speimen design
To haraterize the mehanial behavior of materials subjeted to biaxial loadings,
various ruiform speimens have been dened. Hannon and Tiernan have published
a omplete review on this subjet [21℄. Several researhers ([22℄,[9℄) have proposed a
ruiform speimen shape with a redued thikness square entral zone and grooves in
the arms. An example of suh geometry isshown onFigure1.
Figure1. Geometryof Speimen1.
To study the inuene of biaxial loadings on hardening anisotropi materials,
Ferron andMakinde [20℄ and Demmerleand Boehler [23℄have optimizedthis speimen
and obtained an homogeneous stress-strain eld over a large part of the square zone.
Johnston et al. [24℄ modied the speimen 1 and added a irular entral zone with a
smaller thikness (Figure2).
Figure2. Geometryof Speimen2.
Removing arm grooves, Yong et al. [25℄ proposed the shape shown Figure 3.
by numerial nite element simulations. The initial speimen dimensions dened by
the authors lead to a very sti speimen: the arm width is 90mm and the entral
area diameter is 20mm. But to date, to our knowledge, no experimental results were
published to validate this shape.
Figure3. Geometryof Speimen3.
Anotherruiformspeimenshape(Figure4)withareduedthiknessintheentral
area and variable width arms has been proposed by Zhang and Sakane [26℄. This
speimen has been used torealize fatigue tests, i.e. atlowstrain levels.
Figure4. Geometryof Speimen4.
2.1. Numerialinvestigations
Through FE simulations, the eÆieny of the four various speimen shapes mentioned
above is numerially investigated. For that, equi-biaxial tensile test is simulated by
meansoftheFEsoftwarepakageABAQUS.Consideringthesymmetryofthespeimen
elements are used and a rened mesh is adopted in the sensitive areas where strain
loalizations ould our (entral zone, intermediate setion, llet, grooves). The
isotropi elasto-plasti behavior of an aluminium alloy is assumed. The elasti part
isdesribed byHooke's modelwith Young modulus,E =70000MPaand Poissonratio,
=0:3. Fortheplastipart,isotropivonMisesyieldriteriaisusedandthehardening
behavior isdesribed by a Ludwik's law:
=
0 +K"
n
(1)
whereand"aretheequivalentstressandtheequivalentplastistrainrespetively,
0
istheYieldstressobtainedfromamono-axialtensiletest,K =553MPaandn=0:61
are materialparameters.
The variousshapes suggested (gures 1to 4)were dimensionedwith the following
rules: (i)theharateristidimensionoftheentralzoneisxedatavaluerangingfrom
20to30mm. Thisdimensionismainlyimposedbythedynamiaquisitionapaitiesof
the amera, (ii)other prinipaldimensions (widthand thikness ofthe arms, thikness
of the entral area) are hosen aording to either the maximum load apaities of
the experimental setup or the maximum stiness of the sample. The maximum value
of the stiness is dened by the apaities of the experimental devie. Consequently,
the thikness of the entral area and arms are respetively xed to 1 and 4mm. For
speimens 2to 4,the thikness of the intermediate zone is 2mm.
2.2. Results and disussion
To ensure that neking always appears inthe entralzone, the design of the ruiform
speimens must indue the greatest deformations in the entral zone and no strain
loalizations in the other areas (grooves, llets, ...). Both the equivalent plasti strain
and the maximal prinipal strain in the speimen are observed. In the geometrial
singularities(groovesforexample),theobservationofthe equivalentplastistrainisnot
suÆient,sinethemajorprinipalstrainan reahhighvaluesleadingtoaloalization
of the deformation(and then frature) although the equivalent strain is not very high.
The gures 5 and 6 show the distribution of the equivalent plasti strain (PEEQ) and
the maximal prinipal strain elds for speimens 1 to 4. Only the interesting zone,
i.e. the entral zone is observed. All the results presented orrespond to the moment
whenthemaximumvalueoftheequivalentplastistrainreahes30%inonepointofthe
speimen. Inordertoevaluatetheinueneoftheshapedetailssuhasthegrooves, the
lletofthe arms,the evolutionofthestrains(equivalentplastiandmaximalprinipal)
is alsoplottedalong a path (represented by a mixed line).
Aording to gures 5 and 6, one an see that the strain onentration is more
homogeneousinthe entraltest setionofthespeimenswithgrooves(speimens1and
2). But, forthe speimen1,the maximumstrain value (30%)isreahed inthe grooves.
This phenomenon is less pronouned for speimen 2 where an intermediate zone has
been added. The equivalent plasti strain is then slightly higher in the entral zone
Figure5. Equivalentplasti strainandmaximalprinipalstraineldsforspeimens
1and2.
than in the grooves. But, the maximal prinipal strain is always muh higher (27:5%)
in grooves than in the entral area (15%). For the speimen 3, the maximal prinipal
strain is obtained in the llet of the arms (25%) whereas in the entral zone the level
reahed is approximately 15%. For the speimen 4, the highest strain values (maximal
prinipalstrainorequivalent plastistrain)are loatedinthe lletofthe arms whereas
in the entralarea, the value of the maximum prinipalstrain is the lowest one among
the four speimens(11%).
Table1. Stinesses ofthetestedspeimens.
Speimen 1 2 3 4
Stiness (KN/mm) 42 45 62 38
The results of the FE simulations show that the speimens 2 and 3 seem to be
the more eÆient sine they lead tothe maximum values of strain in the entral zone.
But speimen 3 is, for all the speimens tested, the stiest (Tab. 1). Moreover, for all
the speimens, to redue the stress/strain onentrations in the llets of the arms, it
wouldbeadvisabletoinreasethe widthofthearmsbut thatwouldresultininreasing
signiantly the stiness of the speimen. Finally, the shape of speimen proposed
gure 2 seems tobe the most eetiveand the most promising. The positioningof the
Figure6. Equivalentplasti strainandmaximalprinipalstraineldsforspeimens
3and4.
be performed, FE simulationsshow notably that the distane between the ends of the
grooves and the edge of the square entral zone, had a great inuene on the level of
deformationinthe groovesand lletsofthe arms. Thereafter,onthe basis ofthe shape
ofthesample2,amodiedformisproposed andoptimizedinordertoreduethestrain
loalizationin grooves.
3. Speimen optimization
The speimen hosen inthe previouspart must be optimized inorder tomakeeÆient
its use for a whole forming limit urve identiation. Finally, the validity of the
optimization is veried with the determinationof a numerialFLC for the above used
material.
3.1. Optimization of the ruiform speimenshape
The main purpose of the optimization is to ensure a strain loalizationat the entral
pointof the speimenand thenthe onset ofnekinginthis zone. The strainpath value
at the entralpoint of the speimen is diretly linked tothe veloity ratio of atuators
for the two tested axes. Ifthe strain loalizationappears ina dierent zone, the strain
pathinthis zonewillbedierentfromthe veloityratioandthenonlyapartialforming
limiturve willbe drawn. Thereafter on the basis of the shape illustrated by gure 7,
a modiation of the formof the entral zone isproposed.
Figure7. Geometryof theoptimizedspeimenshape.
Aording to the previous study, two key geometrial parameters have been
identied for the strain distribution in the speimen. These parameters are the length
L between the ends of the grooves and the edge of the square entral zone and the
llet of arms R
(Figure7). As seen before, by adjusting these parameters toadequate
values, the strain level in grooves and llets of the arms is limitedby omparison with
the levelin the entralregionof the speimen. This isthe rst step of the optimization
proedure, the seond step onsistsin the strain loalizationatthe entralpoint of the
speimen for the strain path ontrol. Hereafter, toahieve the proedure, two dierent
shapes are proposed and disussed for the entral zone of the speimen. For speimen
I, a hamfer between the at entral irular zone and the intermediate square zone is
dened (Figure 8) and for speimen II, the entral point and the intermediate square
zone are onneted by a urved prole(radius) (Figure9).
Figure8. SpeimenI:transition withhamfer.
One an see in gures 8 and 9 that the thikness of the entral test setion T is
also a key parameter for the strain loalisation and homogeneity. For speimen I, the
