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Theoretical and numerical study of strain rate influence on AA5083 formability
Cunsheng Zhang, Lionel Leotoing, Dominique Guines, Eric Ragneau
To cite this version:
Cunsheng Zhang, Lionel Leotoing, Dominique Guines, Eric Ragneau. Theoretical and numerical study
of strain rate influence on AA5083 formability. Journal of Materials Processing Technology, Elsevier,
2009, 209, pp.3849-3858. �10.1016/j.jmatprotec.2008.09.003�. �hal-00983355�
inuene on AA5083 formability
Cunsheng ZHANG,Lionel LEOTOING, Dominique GUINES,EriRAGNEAU
INSA,Laboratoire deGénie Civil etGénie Méanique (LGCGM, EA 3913)
20 Av. des Buttes deCoësmes, 35043 RennesCedex, Frane
Email:lionel.leotoinginsa-rennes.fr
Tel :+33(0)2 2323 8664 Fax: +33(0)2 2323 8726
Abstrat
With the appliation of new forming tehniques (hydroforming, inremental form-
ing), it is neessary to improve the haraterization of theformabilityof materials
and in partiular the inuene of strain rate. This paper begins with the hara-
terizationofmaterialbehaviorofan aluminumalloy5083at hightemperatures. To
desribeitsviso-plastibehavior,Swift'shardeninglawisusedandtheorrespond-
ing parameter values are identied. Then, two dierent approahes are introdued
to onstrut FLDs(forminglimit diagrams) ofthis alloysheet and evaluate theef-
fetof the rate-sensitivity index on its formability.The rst one is theoretial (the
M-K model), and an algorithm is developed to alulate the limit strains by this
model. In the seond approah, the Mariniak test is simulated with the ommer-
ially available nite-element program ABAQUS. Based on FEM results, dierent
failureriteriaaredisussedandanappropriateoneishosentodeterminetheonset
of loalized neking.With the material behavior data orresponding to AA5083at
150 Æ
C, parametri studies are arried out to evaluate the eet of the strain rate
sensitivityindex.Theomparisonofresultsbythesetwoapproahesshowsthesame
tendeny thatan improvement of the formability with inreasing strain rate sensi-
tivityisobserved.Finally,byonsiderationoftheompensatingeetsofthestrain
hardening and ratesensitivityindies, theFLDsof this sheetat 150 C,240 C and
300 Æ
C aredetermined andompared. Resultsshowthat theformabilityof AA5083
seemsnottobeimproved uptoaertaintemperature (between 240 Æ
C and300 Æ
C),
above this temperature, theformability isgreatly enhaned.
Key words: FormingLimit Diagrams(FLDs);Mariniak test; strainrate
sensitivity
1 Introdution
Thesheetmetalformingreeivesmoreandmoreappliationinthedomainsof
automotive and aeronautis. Espeially with the innovative tehniques, suh
as hydroforming and inremental forming, the manufature of omplexparts
with low tools ost an be realized. These proesses are generally performed
in the intermediate range of strain rates (10 2
to 500s 1
). However, insheet
metal forming operations, the sheet an be deformed only to a ertain limit
that is usually imposed by the onset of loalized neking, whih eventually
leadstofrature.Awell-knownmethodof desribingthislimitand prediting
the ourrene of neking is the forming limit diagrams (FLDs) introdued
by Keeler and Bakofen in the 1960s [Keeler and Bakofen(1963)℄. In FLDs,
a FLC (forming limit urve) represents a plot of major and minor available
prinipalstrains in the plane of the deformed sheet orresponding to the o-
urrene of the neking.
The determination of FLDs is a omplex task, and researh on FLDs has
always been the subjet of extensive experimental, theoretial and numerial
studies. Forexperimental determinationof FLDs, two main kindsof forming
methods have been developed, the so-alled out-of-planestrething (e.g., the
Nakazimatest,theHekertest)andthein-planestrething(e.g.,theMariniak
test). Byforming a number of sheet speimenswith varying widths,dierent
strainstateisevaluatedjustoutsidethefraturezonebytheirlegridmethod
orthe digitalimage orrelationtehnique. Finally,by onneting allthe limit
strain points,the FLC isdrawn.
To eetively study plasti instability phenomenon and simplify the deter-
mination of FLDs, researh has been mainly foused on development of the
mathematial models for theoretial determination of FLDs. As early as in
1952, Swift[Swift(1952)℄ developed ariterion forprediting the onset of dif-
fuse neking with the assumption that plasti instability ours at a maxi-
mum load. However, in industrial stampings, the maximum allowable strain
is determined by the loalized neking rather than by diuse neking. Hill
[Hill(1952)℄ proposed a loalized neking riterion based on the well-known
zero extension assumption (for a negative minor strain), i.e., the loalization
band develops normal to the diretionof zero extension in a sheet metal.On
the basis of the experimentalinvestigations onerning the strainloalization
of some speimens subjeted to biaxialstrething, Mariniak and Kuzynski
[Mariniak and Kuzynski(1967) ℄introduedin1967imperfetionsintosheets
toallowneking totakeplae (known as the M-Kmodel).The imperfetions
an be aused by fators suh as loal grain size variation, texture, alloys el-
ements, thikness variation,et. Today, the M-Kmodelhas been widely used
topreditFLDs,and the originalM-Kmethodhas undergonegreat improve-
ment.
With inreasing appliation of omputational tehniques, numerial pre-
ditions of FLDs have beome more attrative and the nite element
method (FEM) has been seleted to simulate the Nakazima and Mariniak
tests. In analyzing the simulation results for the onset of neking, it
is essential to establish a failure riterion. One of the pioneers was
Brun [Brun etal.(1999)Brun, Chambard, Lai, and De Lua℄ who has ana-
lyzed thinning of sheets in order to determine the onset of neking by the
Basing on the same test, Geiger and Merklein [Geiger and Merklein(2003)℄
onsidered that the gradient of major strain hanged rapidly when lo-
alized neking ourred. Using the limiting dome height (LDH) test,
Narasimhan [Narasimhan(2004)℄ has predited the onset of neking by
the thikness strain gradient aross neighboring regions. Additionally,
the LDH test was arried out by Zadpoor et al. with ABAQUS into
whih an improved M-K model with Stören-Rie's analysis was imple-
mented [Zadpooret al.(2007)Zadpoor,Sinke, and Beneditus ℄. Predited re-
sultsshowed that whilethe originalM-Kmodelonsiderablymisspreditsthe
limitstrains, aombinationof theM-KmodelandStören-Rie's analysisan
predit the dome height with good auray. Based on the Mariniak test,
Petek et al. [Petek etal.(2005)Petek, Pepelnjak,and Kuzman℄ put forward a
new method for the evaluation of the thikness strain as a funtion of time
as well as the rst and seond time derivative of the thikness strain. They
proposed that the maximum of the seond temporal derivative of thikness
strain orresponds to the onset of neking.Volk [Volk(2006)℄ proposed anew
approah for identifying the onset of loalized neking by experimental and
numerialmethods.Withalulatedstrainrates,theidentiationwasarried
out with the two following main eets: inrease of points number with high
strain rate (in the loalization area) and derease of the strain rate outside
the loalization bands. From the above literature, it is observed that FLDs
stronglydepend onthe riteriahosen,therefore,anappropriatefailurerite-
rion is akey tonumerialdeterminationof FLDs.
AlthoughFLDs have been suessfully used and proved tobe apowerfultool
in sheet metal forming analysis, there are still shortomings to be overome.
Firstly,tothisday,thereisnotpreisestandardforthedeterminationofFLDs.
Moreover, ithas beenfoundthat suhforminglimitshangesigniantlywith
alterations in the strain path [Arrieux(1990)℄. To remove this limitation, a
Additionally, relatively little attention has been paid to the models of FLDs
takingthestrainratesensitivityintoaount.Strainhardeningandstrainrate
sensitivity have been identied as important fators for determining forma-
bility of sheet metal and alter substantially the level and shape of FLCs.
Experimentally,Laukonis and Ghosh [Laukonis and Ghosh(1978)℄ found that
strain rate eet is very sensitive for AKsteel, espeiallyfor the deformation
modenearbiaxialstrething,whilealuminumseemstobeinsensitivetostrain
rate.Pery[Pery(1980) ℄analyzedthe inueneof strainrateonFLDsby ex-
plosive forming and onluded that FLDs level was dependent on the strain
stateandformingrates.Broomheadetal.[Broomhead and Grieve(1982)℄per-
formedbulgeformingoverarangeof strainratesfrom10 3
to70s 1
andon-
luded that the position of FLDs under biaxial tensile onditions dereased
with inreasing strain rate. These ontraditory experimental results under-
linethe diulty indetermining the onset of neking in the ase of dynami
experiments.Hene,itisneessarytoestablisharigorousproedureandarry
outmoreexperimentalinvestigationsaboutformingbehavioratorresponding
strain rates.
Theoretially, researh on the rate sensitivity on FLDs has been ar-
ried out by several authors using the M-K model. Huthinson et
al. [Huthinson et al.(1978)Huthinson,Neale, and Needleman℄ predited the
FLDs with von Mises' yield funtion taking rate sensitivity into
aount. Their work has given important ontributions to the in-
sight into the inuene of onstitutive equations and plastiity the-
ories on FLDs. Lee and Zaverl [Lee and Zaverl(1982)℄ omputed en-
tire FLD based on the rate-dependent ow theory under proportional
loading with the assumption of zero extension. Barata Da Roha et
al. [Barata da Rohaet al.(1984-1985)Baratada Roha, Barlat,and Jalinier℄
predited thestrain path-dependentFLDsby onsidering ratesensitivityand
ulatedFLDs forrate sensitive materialsby applyingthe isotropi hardening
model of the ow theory for the anisotropi sheet metals. Graf and Hosford
[Grafand Hosford(1990)℄ analyzed the eet of rate sensitivity on the right-
hand side of FLDs with the Logan's and Hosfords' anisotropy yieldriterion.
Today, for the right-hand side of the FLDs, the analysis has been quite su-
essful, whereas to the left-hand side, beause of the omplex algorithmsand
lengthy alulations, relativelylittleattention has been paid.
Therefore,FLD'sstandardizeddetermination,itsnewrepresentations,itssen-
sitives tostrain pathsand strain rate are stilltoday's researhpoints.Exper-
imental results by tensile test at elevated temperatures (150 Æ
C, 240 Æ
C and
300 Æ
C) show a strain rate dependene of aluminum alloy 5083 on tempera-
ture.Inthispaper,weareinterestedintheeetofthisstrainratedependene
onitsformability.This eet isinvestigated by theoretialand numerialap-
proahes. Firstly,analgorithmisdeveloped toalulatethe limitstrains with
theM-Kmodel.Then,the Mariniaktestissimulatedforthis rate-dependent
material with the ommerially available nite-element program ABAQUS.
Finally, based on the above two methods, the eet of rate sensitivity index
onformabilityisevaluatedandFLDsofAA5083sheetatvarioustemperatures
are determined.
2 Strain rate sensitivity of aluminum alloy 5083
Inreased interest in the prodution of lightweight vehiles to improve
fuel eonomy has resulted in an interest in utilization of aluminum al-
loys. In partiular, beause of its relatively good formability and orro-
sion resistane, the aluminum-magnesiumalloy 5083 reeives more and more
appliation in automotive and aerospae industry. Previous studies have
shown the strain rate dependene of the alloy at elevated temperature
the multipliativeSwiftlaw
=K("
0 +")
n
_
"
m
(1)
has been hosen to desribe the viso-plasti behavior of this AA5083 alloy,
where " and _
" are the equivalent plasti strain and the equivalent plasti
strain rate, respetively. Here, n and m are the strain hardening and strain
rate sensitivity indies,and K and "
0
are materialparameters.
ToharaterizethehightemperaturedeformationbehaviorofAA5083,tensile
tests have been performed on a high-speed servo-hydrauli testing mahine
(DARTEC, 20kN apaity) at temperatures of 150 Æ
C, 240 Æ
C and 300 Æ
C and
the onstant rosshead speeds of 1.56, 15.6 and 156 mm/s (orresponding to
intermediatestrainrates fromapproximately10 2
up to10s 1
),respetively.
By the tensile tests, the true stress-true strain urves at 150 Æ
C, 240 Æ
C and
300 Æ
C are obtainedasshown inFig.1, Fig.2andFig.3, respetively.Withthe
least squares method, the orresponding parameter values of above onstitu-
tive materialmodelhave been identied tot experimentaldata as shown in
Tab.1. Here, K, n and m are onsidered to be onstant for a given temper-
ature and m is determined basing on the stress-strain urves with the three
speedsatthistemperature.OneanobservethatAA5083exhibitslittlestrain
ratesensitivityat150 Æ
C (m=0:0068),whilethis sensitivity learlyaugments
with inrease of temperature. On the ontrary, with inreasing temperature
the work hardening index n dereases. The orresponding tting urves are
ompared with experimental stress-strain urves in Fig.1, Fig.2 and Fig.3,
respetively.
At 150 Æ
C, it is observed that the urves identied with Swift's law are in
good agreement with experimental data. On the ontrary, at higher tem-
peratures, some divergenes between experimental and identied urves an
deline from the peak as strain proeeds, while Swift's tting urves al-
ways give inreasing trends. The delination of ow stress with strain af-
ter reahing the peak stress is mainly attributed to material softening
[Lee et al.(2004)Lee,Sohn, Kang, Suh, and Lee ℄. Of partiular interest for us
here is the eet of strain rate sensitivity on the forming apaity of sheets,
therefore,in this work, softening eet is not onsidered.
3 M-K theoretial model
3.1 Brief desription of the M-K model
The typial M-K geometrial model is shown in Fig.4. The imperfetion is
geometriallyrepresented by alonggroovewhihisharaterizedbyaninitial
imperfetionfator
f
0
= e
b
0
e a
0
<1; (2)
where e a
0
and e b
0
are the initial sheet thiknesses in zone a and zone b, and
throughout the analysis the indies a and b are used to designate the zones
outside and insidethe groove,respetively.
In the M-K original model introdued by Mariniak and Kuzyn-
ski [Mariniak and Kuzynski(1967)℄, the groove is perpendiu-
lar to the prinipal stress, i.e.,
0
= 0. Later, Huthinson et al.
[Huthinsonet al.(1978)Huthinson,Neale, and Needleman℄ extended
this model to strain paths in the negative minor strain region based
on a groove inlined at an angle
0
with respet to the prini-
pal axis-2 (Fig.4). They put forward that the limit strains un-
der uniaxial tension varied with initial groove orientation, as well
[Barata daRohaet al.(1984-1985)Baratada Roha, Barlat, and Jalinier℄
alsoonluded that in most ases the ritial strains were ahieved for initial
groove orientations dierent from zero. For the right side of FLDs, many
researhersahieved ritialstrainswiththe simplistimodel
0
=0.Banabi
andDannenmann [Banabi and Dannenmann(2001)℄appliedHill's1993yield
riterionintheM-Kmodelandanalyzedtheinueneoftheyieldurveshape
upon the right-hand side of FLDs. Avila and Vieira [Avila and Vieira(2003)℄
developed an algorithm for predition of the right-hand side of FLDs based
onthe M-K model. Fivedierentyield riteria(von Mises', Hill's1948, Hill's
1979,Hosford'sandHill's1993)wereimplantedintothisalgorithmtoanalyze
their inuene onFLDs.
Inthe followingwork,the numerialanalysis oftheM-Kmodelforbothases
(
0
= 0 and
0
6=0) is illustrated. For the left-hand side of FLDs, the ase
0
6= 0 (general ase) is onsidered, while
0
=0 (partiular ase) is for the
right-sideof FLDs.
Thankstothesheetplanequasi-isotropyofAA5083,vonMises'syieldfuntion
under plane stress assumption ( k
13
= k
23
= k
33
= 0) an be used to model
this sheet behavior
k
2
=
k
11
2
k
11
k
22 +
k
22
2
+3
k
12
2
; (3)
where k
istheequivalentstress, k
11 ,
k
22 and
k
12
arestresstensoromponents,
k =a (or b).
The sheet metal obeys Levy-Mises' ow rule, whih an be expressed in the
form
"
k
ij
=
k
k
ij
"
k
(i,j=1,2); (4)
where "
k
ij
and "
k
are the strain omponent inrements and the equivalent
plasti strain inrements, respetively, and refers to a hange orrespond-
assumed during this analysis.
In the M-K model, the same fore in the diretion-n (Fig. 4) is transmitted
aross zones a and b. Therefore, the equilibriumequationsare
a
nn e
a
= b
nn e
b
;
a
nt e
a
= b
nt e
b
; (5)
where e a
, e b
are the urrent sheet thiknesses.
Thestraininzoneb, paralleltothegroove,isonstrainedbythe uniformzone
a so that the ompatibilityondition is
"
a
tt
="
b
tt
: (6)
3.2 Partiular ase (positive minor strain)
The initialimperfetionisassumed to beperpendiular tothe prinipalaxis-
1,
0
= 0, in the partiular ase onsidered here. The groove referene and
main axessystem oinidefor both zones. The eqs.(5) and (6) redueto
a
11 e
a
= b
11 e
b
(7)
and
"
a
22
="
b
22
: (8)
Forthe sake of onveniene, the notations
k
=
"
k
22
"
k
11
;
k
=
k
22
k
11
= 2
k
+1
2+ k
' k
=
k
k
11
= q
1
k
+( k
) 2
; k
=
"
k
"
k
11
= 2'
k
2
k
(9)
The equivalent strain rate an be expressed in terms of the strain and time
inrementsas
_
"
k
=
"
k
t
: (10)
WithSwift'shardeninglaw(1),theowrule(4)andtheeqs.(8-10),theeq.(7)
an be expressed as
("
0 +"
a
+ a
"
a
11 )
n ('
a
) m 1
(2 a
1) m
=f
"
0 +"
b
+ b
"
b
11
n
' b
m 1
(2 b
1) m
; (11)
where f is the urrent imperfetion fator. Equation (11) shows that with
the disappearane of time inrement, the level of strain rate has no eet in
theM-Kmodel.Therefore, onlytherate-sensitivityvis-à-vistheparameterm
ould be analyzedfor a given strain rate.
Under the assumption of proportional loading in zone a, the strain path is
haraterized by a onstant strain ratio a
. The parameter "
a
11
is known.
Therefore the terms k
, ' k
and k
are onstant for a ertain a
and an
be easily alulated. For zone b the orresponding quantities vary with the
strain inrements but all an be expressed as funtions of "
b
11
by use of the
ompatibilityondition.
To alulate"
b
11
, the funtion
F
"
b
11
=("
0 +"
a
+ a
"
a
11 )
n 2
b
1
2 a
1
!
m
f
"
0 +"
b
+ b
"
b
11
n
' b
' a
!
m 1
(12)
is used. To numerially solve the equation F
"
b
11
=0, Newton-Raphson's
method isused. The (i+1)th iterationstep is
"
b
11
(i+1)
"
b
11
(i)
= F
"
b
11
( i)
dF=d
"
b
11
( i)
: (13)
When absolute values of the inrement
"
b
11
(i)
beome less than an error
