Prediction of anisotropy and hardening for metallic sheets in tension, simple shear and biaxial tension S.L. Zang a,b,1 , S. Thuillier a,b, , A. Le Port a,b , P.Y. Manach a,b a LIMATB, Universite´ de Bretagne-Sud, Rue de Saint Maude ´, BP 92116, F-56321 Lorient, France b Universite´ europe´enne de Bretagne, France article info Article history: Received 15 May 2010 Received in revised form 4 November 2010 Accepted 9 February 2011 Available online 21 February 2011 Keywords: Anisotropic yield Constitutive behavior Metallic material Mechanical testing abstract The mechanical behavior of mild and dual phase steel sheets is investigated at room temperature in quasi-static conditions under different strain paths: uniaxial tension, simple shear and balanced biaxial tension. The aim is to characterize both the anisotropy and the hardening, in order to identify material parameters of constitutive equations able to reproduce the mechanical behavior. In particular, a good description of flow stress levels in tension and shear as well as plastic anisotropy coefficients is expected. Moreover, the Bauschinger effect is investigated with loading–reloading in the reverse direction shear tests and the balanced biaxial tension test gives insight of the mechanical behavior up to very high equivalent plastic strains. Yoshida–Uemori hardening model associated with Bron–Besson orthotropic yield criterion is used to represent the in-plane mechanical behavior of the two steels. The identification procedure is based on minimization of a cost function defined over the whole database. The presented results show a very good agreement between model predictions and experiments: flow stress during loading and reverse loading as well as plastic anisotropy coefficients are well reproduced; it is shown that the work-hardening stagnation after strain path reversal is well estimated in length but Yoshida–Uemori model underestimates the rate of work-hardening. & 2011 Elsevier Ltd. All rights reserved. 1. Introduction Nowadays, phenomenological models are widely used in finite element analysis of sheet metal forming process, since they present a good compromise between simulation accuracy and computation time. Such models of the elasto-viscoplastic behavior of sheet metals are based on the definition of a yield surface, to describe the initial anisotropy related to the crystallographic texture, and its evolution with plastic strain. Initial orthotropy is a good represen- tation for rolled sheets and is assumed to be kept during strain, by considering a corotation of the anisotropy frame with material rotation. Strain-induced anisotropy, such as Bauschinger effect, is described by the evolution of internal variables with plastic strain. Several experimental tests, like tension-compression [1,2] and simple shear [3], have been performed to characterize the hard- ening behavior of sheet metals under strain reversal, which refers to the fact that the subsequent loading direction is opposite to that of former loading, and is quite common in sheet metal forming processes; for example, bending–unbending on the die radius and reverse bending–unbending at the punch nose. This behavior under strain reversal, called the Bauschinger effect, is characterized by a lower yield stress under strain reversal, further transient behavior that corresponds to the smooth elastic-plastic transition with a rapid change of strain-hardening rate, and a hardening stagnation, the magnitude of which depends on the prestrain and permanent softening characterized by stress offset. The prediction of the anisotropy and hardening of metallic sheets depends not only on the constitutive model but also on the accurate material parameter identification which refers both to the type of the experimental tests being used and the identifica- tion methods. Tension, simple shear and balanced biaxial tension tests provide relevant information on the shape of the yield surface and its evolution with plastic strain. However, current researches seldom consider all of them to identify the material parameters. The general identification strategy is that the first step is the identification of the initial yield surface, using either the yield stresses or the anisotropy coefficients, or both, and the second step is the hardening behavior, e.g. [4]. In the present study, an alternative procedure is used and the material parameters of both the yield function and the hardening model are identified from the stress–strain curves and both longitudinal and transverse strain in tension at the same time. The constitutive equations are derived from Bron–Besson yield Contents lists available at ScienceDirect journal homepage: www.elsevier.com/locate/ijmecsci International Journal of Mechanical Sciences 0020-7403/$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijmecsci.2011.02.003 Corresponding author at: LIMATB, Universite ´ de Bretagne-Sud, Rue de Saint Maude ´ , BP 92116, F-56321 Lorient, France. Tel.: + 33 297874570; fax: + 33 297874572. E-mail address: [email protected] (S. Thuillier). 1 Present address: School of Mechanical Engineering, Xi’an Jiaotong Univer- sity, No. 28, Xianning Road, Xi’an, Shaanxi, China. International Journal of Mechanical Sciences 53 (2011) 338–347
Transcript
Prediction of anisotropy and hardening for metallic sheets in tension, simpleshear and biaxial tension
S.L. Zang a,b,1, S. Thuillier a,b,!, A. Le Port a,b, P.Y. Manach a,b
a LIMATB, Universite de Bretagne-Sud, Rue de Saint Maude, BP 92116, F-56321 Lorient, Franceb Universite europeenne de Bretagne, France
a r t i c l e i n f o
Article history:Received 15 May 2010Received in revised form4 November 2010Accepted 9 February 2011Available online 21 February 2011
The mechanical behavior of mild and dual phase steel sheets is investigated at room temperature inquasi-static conditions under different strain paths: uniaxial tension, simple shear and balanced biaxialtension. The aim is to characterize both the anisotropy and the hardening, in order to identify materialparameters of constitutive equations able to reproduce the mechanical behavior. In particular, a gooddescription of flow stress levels in tension and shear as well as plastic anisotropy coefficients isexpected. Moreover, the Bauschinger effect is investigated with loading–reloading in the reversedirection shear tests and the balanced biaxial tension test gives insight of the mechanical behavior upto very high equivalent plastic strains. Yoshida–Uemori hardening model associated with Bron–Bessonorthotropic yield criterion is used to represent the in-plane mechanical behavior of the two steels. Theidentification procedure is based on minimization of a cost function defined over the whole database.The presented results show a very good agreement between model predictions and experiments: flowstress during loading and reverse loading as well as plastic anisotropy coefficients are well reproduced;it is shown that the work-hardening stagnation after strain path reversal is well estimated in length butYoshida–Uemori model underestimates the rate of work-hardening.
& 2011 Elsevier Ltd. All rights reserved.
1. Introduction
Nowadays, phenomenological models are widely used in finiteelement analysis of sheet metal forming process, since they presenta good compromise between simulation accuracy and computationtime. Such models of the elasto-viscoplastic behavior of sheetmetals are based on the definition of a yield surface, to describethe initial anisotropy related to the crystallographic texture, and itsevolution with plastic strain. Initial orthotropy is a good represen-tation for rolled sheets and is assumed to be kept during strain, byconsidering a corotation of the anisotropy frame with materialrotation. Strain-induced anisotropy, such as Bauschinger effect, isdescribed by the evolution of internal variables with plastic strain.Several experimental tests, like tension-compression [1,2] andsimple shear [3], have been performed to characterize the hard-ening behavior of sheet metals under strain reversal, which refersto the fact that the subsequent loading direction is opposite to thatof former loading, and is quite common in sheet metal forming
processes; for example, bending–unbending on the die radius andreverse bending–unbending at the punch nose. This behaviorunder strain reversal, called the Bauschinger effect, is characterizedby a lower yield stress under strain reversal, further transientbehavior that corresponds to the smooth elastic-plastic transitionwith a rapid change of strain-hardening rate, and a hardeningstagnation, the magnitude of which depends on the prestrain andpermanent softening characterized by stress offset.
The prediction of the anisotropy and hardening of metallicsheets depends not only on the constitutive model but also on theaccurate material parameter identification which refers both tothe type of the experimental tests being used and the identifica-tion methods. Tension, simple shear and balanced biaxial tensiontests provide relevant information on the shape of the yieldsurface and its evolution with plastic strain. However, currentresearches seldom consider all of them to identify the materialparameters. The general identification strategy is that the firststep is the identification of the initial yield surface, using eitherthe yield stresses or the anisotropy coefficients, or both, and thesecond step is the hardening behavior, e.g. [4].
In the present study, an alternative procedure is used and thematerial parameters of both the yield function and the hardeningmodel are identified from the stress–strain curves and bothlongitudinal and transverse strain in tension at the same time.The constitutive equations are derived from Bron–Besson yield
function [5] and Yoshida–Uemori hardening model [6] andidentification of material parameters in tension, monotonic andBauschinger simple shear and balanced biaxial tension tests arepresented for DP500 and DC04 steels: balanced biaxial tensionallows to reach high equivalent plastic strain, whereas simpleshear involves large material rotations. Considering these threetests leads to a complementary experimental database well suitedfor phenomenological approach, though it is rather constrainingfor parameter identification. The ability of Yoshida–Uemori modelto predict the work-hardening stagnation after strain path rever-sal in simple shear is particularly studied. This test allows anequivalent plastic prestrain up to 0.2 to be investigated and toreach after straining an equivalent plastic strain of 0.5.
2. Experiments
Two thin sheet materials are considered in this study: a mildsteel DC04, with a thickness e! 0:67 mm and a dual phase steelwith a thickness of 0.6 mm and a tensile strength of 500 MPa(DP500). For this last material, SEMmicrographs have evidenced avolume fraction of martensite around 6% and a grain size of 5 mm.The mechanical behavior of these two steels is investigated underthree different stress and strain states, i.e. uniaxial tension, simpleshear (both of these tests are performed at several orientations tothe rolling direction or RD) and balanced biaxial tension. Theexperimental procedure is described in the following paragraphs.
2.1. Tension
Tensile tests were carried out on rectangular samples ofdimension 20 "180 " e mm3. The free edges were machined inorder to eliminate the hardened areas induced by the cutting andthus to increase the range of homogeneous deformation. Compo-nents of the strain tensor in the sheet plane are calculated by
image correlation. Monotonous tensile tests were carried out at01, 451 and 901 to the RD for DP500 and in addition at 221 and 771for DC04, in order to study the material anisotropy. For thesetests, a cross-head speed of 10 mm/min is imposed which leads to_eC2:4" 10#3 s#1. The logarithmic strain as well as the Cauchystress are calculated from the raw data (Fig. 1). The plasticanisotropy coefficients ra ! depYY=de
pZZ , where ~eX denotes the ten-
sile, ~eY the transverse and ~eZ the normal directions, respectively,and a the angle between the RD and the tensile direction, arecalculated from the transverse strain eYY and the assumption ofvolume conservation in the plastic area; they are given in Table 1.
The deformation gradient F [7] in the central zone of thesample is given by F ! FXX~eX $~eX%FYY~eY $~eY %FZZ~eZ $~eZ where~ei,i! X,Y ,Z are the basis vectors of the global reference frame.The test is controlled by the evolution of FXX with time and byconstraining sYY ! sZZ ! 0. The signals calculated from therecorded raw data are the components FYY and sXX ! load/(actualsection). The strain range is limited to its maximum value beforenecking, which corresponds to 0.18 for DP500 and 0.25 for DCO4,whatever the orientation to RD.
2.2. Simple shear
The simple shear device is presented in detail in [8]. Thespecimens have a rectangular shape, a gauge area of lengthL ! 50 mm and width h of 4 mm; the shear direction is alongthe length of the specimen (Fig. 2). The samples are kept underthe grips by six screws tightened by a dynamometric key whichtorque is calibrated depending on the tested material. Theoptimal value is obtained with the lowest torque that minimizesthe sliding between the sample and the grips. Monotonous sheartests were performed on samples at the same orientations to theRD than for the tensile test, at a cross-head speed of 0.5 mm/min,which corresponds to _g ! 2:1" 10#3 s#1. Moreover, cyclic testsare performed in order to highlight the Bauschinger effect and tomeasure kinematic hardening parameters. These tests are com-posed of a loading up to several values of g followed by a load inthe opposite direction until g!#0:4. Each kind of test isperformed three times to check the reproducibility and a repre-sentative test is chosen for the database. Shear strain g, whichcorresponds to the non-diagonal component of the planar trans-formation gradient in the case of an ideal simple shear kine-matics [8], is measured from a digital correlation system and isthen defined as an average over a rectangular zone on the samplesurface. Fig. 2 shows a rather constant value of g except near thefree ends of the specimen, over a distance of approximately 5 mm.
The kinematics of the simple shear test can be described byF ! I%FXY~eX $~eY with I the second order identity tensor. The testis controlled by the evolution of FXY with time, where~eX is parallelto the shear direction and ~eY perpendicular to ~eX in the sheetplane, and by constraining siZ ! 0&i! X,Y ,Z'. This assumption of aplanar stress state comes from the small sheet thickness.
2.3. Balanced biaxial tension
A hydraulic bulge test, developed in the Universite deBretagne-Sud (A. Penin, V. Grolleau, unpublished results, 2001),
Fig. 1. Cauchy stress versus logarithmic longitudinal strain evolution for DP500 intension. The strain range is investigated a little further necking, evidenced on thenominal stress–strain curve, and by neglecting any triaxiality effects in this area.This assumption has been validated by finite element simulation.
Table 1Plastic anisotropy coefficients of the two steels. The average anisotropy coefficient r ! &r0%r90%2r45'=4, which characterizes the normal anisotropy and the planaranisotropy, measured by the coefficient Dr! &r0%r90#2r45'=2 are also given.
S.L. Zang et al. / International Journal of Mechanical Sciences 53 (2011) 338–347 339
is used to obtain a balanced biaxial strain state. Circular blanks ofgauge diameter of 185 mm are clamped by screws between ablank-holder and a die with a radius of 8 mm. A fixed volume ofwater is pressed under the blank by the displacement of anactuator. A pressure sensor gives the fluid pressure and the strainfield is measured by digital correlation in an area around thecenter point (Fig. 3(a)). The strain state on the surface is recordedduring the test; an average of the strain components over a smallarea around the specimen center is performed and the evolutionof eXX and eYY with time is compared (Fig. 3(b)), with~eX parallel tothe RD and ~eY its perpendicular in the sheet plane. It can be
shown that the two components are very close to each other forboth materials, though only one (DC04) is presented here. Therelative gap &eXX#eYY '=eXX is lower than 4%, for strains above 0.05.The curvature radius RBT is determined by fitting a sphere over theselected area, which leads to an average value. A balanced biaxialstress state is assumed and its non-zero component is calculatedfrom the bulging pressure P and the current blank thickness e:
sb !PRBT
2e&1'
As shown in Fig. 3(a), eXX ( eYY and, therefore, a balancedbiaxial strain state is imposed. In the identification procedure, thebulge test is considered as a homogeneous test withF ! FXX&~eX $~eX%~eY $~eY '%FZZ~eZ $~eZ . The test is controlled bythe evolution of FXX with time, and the stress tensor is given byr! sb&~eX $~eX%~eY $~eY '. The strain range is limited to its max-imum value before localization, evidenced by the pressuredecrease, which corresponds to 0.3 for both DP500 and DCO4.
3. Constitutive equations
In this paper, the material model is a modification of the elasto-plastic model of [6] in which Bron–Besson non-quadratic ortho-tropic yield function [5], a non-saturating isotropic strain-hard-ening and the viscous character of the material are taken intoaccount. Moreover, constitutive equations are written in theanisotropy frame {RD,TD,ND}, the orientation of which is constantduring deformation compared to the corotational frame. Aftereach increment, the tensorial variables are rotated back to thecorotational frame and then to the global reference frame. In thefollowing, the corotated strain tensor e, elastic strain ee, viscoplas-tic strain evp and Cauchy stress s in the anisotropy frame are used.
3.1. Hardening model
In elasto-viscoplastic constitutive model, the rate of s can bewritten as
_s ! C : &_e#_evp' &2'
where C is the elasticity modulus tensor.The viscoplastic strain tensor follows a flow rule derived from
a viscoplastic potential O which is a power function of overstress
0
0.15
0.3
0.45
0.6
0 15 30 45
#
X
S0S1S2
average 1average 2
50 m
m
4 mm
Fig. 2. Homogeneity of the strain distribution along three sections (S0, S1, S2) parallel to the shear direction, for g! 0:3 and 0:6. X is along the sample length. Either theentire gauge surface is used (average 1) or a reduced area in the specimen center (average 2). In the following, the second measure is used.
0
2
4
6
0 0.15 0.3 0.45
P (M
Pa)
"xx
IF04-1IF04-2
DP500-1DP500-2
0
0.15
0.3
0.45
0 100 200-10
-5
0
"
rela
tive
gap
(%)
time (s)
"XX"YY
relative gap
185 mm
Fig. 3. Biaxial expansion. (a) Evolution of pressure P versus eXX in balanced biaxialexpansion. Two tests for each material are plotted to show the good reproduci-bility obtained. (b) Comparison of eXX and eYY for material DC04. The relative gap isdefined by &eXX#eYY '=eXX .
S.L. Zang et al. / International Journal of Mechanical Sciences 53 (2011) 338–347340
sv ! f% 40 or viscous stress [9]:
O&f ' !Kv
nv%1f%Kv
! "nv %1
&3'
where nv is the strain rate sensitivity coefficient and Kv aweighting coefficient of the viscous part of the stress. Thebehavior is thus elastic if fr0 and if f40, the viscoplastic strainrate is written as
_evp !@O@s
!O0&f '@f@s
&4'
The equivalent viscoplastic strain rate _p is defined by
_p !
#####################23_evp : _evp
r&5'
The model proposed by Yoshida and Uemori [6] is constructedwithin the framework of two-surface modeling [10]; only kine-matic hardening of the yield surface f is assumed, but mixedisotropic-kinematic hardening for the bounding surface G:
f !f&s,a'#Y ! 0 &6'
where f is a scalar function measuring the size of the yieldsurface, the second order tensor a is a back-stress representingthe center of the yield surface, and Y is fixed and represents theinitial yield stress.
The bounding surface is described by the equation:
G!f&R,b'#&B%R' ! 0 &7'
where R is a stress tensor, b denotes the center of the boundingsurface, and B and R are, respectively, the initial size and theisotropic hardening (IH) component of the bounding surface.
The kinematic hardening of the yield surface is calculatedfrom:
a! X%b &8'
Evolution of back-stress tensor X is defined as
_X ! CaY
$ %&s#a'#
####a
X
rX
& '_p with X !f&X' and a! B%R#Y
&9'
where C is a material parameter that controls the rate of thekinematic hardening.
The following evolution equation is assumed to describe thekinematic hardening of the bounding surface:
_b !m23b_evp#b _p
! "&10'
where m and b denote material parameters.In the present paper, the Swift law is used to describe the non-
saturating strain-hardening for some materials within certainrange of large strain instead of Voce law [11]:
_R ! nK&p%e0'n#1 _p with e0 !BK
! "1=n
&11'
where K is a material parameter and n the hardening coefficient.Hardening stagnation recorded after strain path reversal is
modeled by the non-isotropic hardening (non-IH) of the boundingsurface; a non-IH surface, gs, is defined in the stress space:
gs&s,q,r' !c&s#q'#r!
################################32&s#q' : &s#q'
r#r! 0 &12'
where q and r denote the center and size of the non-IH surface,respectively. The center of the bounding surface, b, exists eitheron or inside the surface gs. The isotropic hardening of thebounding surface takes place only when the center point of thebounding surface, b, lies on the surface gs, namely _R40 when
gs&b,q,r' ! 0 &13a'
@gs&b,q,r'@b
: _b40 &13b'
otherwise
_R ! 0 &14'
A kinematic motion of the surface gs such that the center of gsmoves in the direction of &b#q' is assumed:
_q ! m&b#q' &15'
From the consistency condition that the center of the bound-ing surface, b, should be either on or inside gs, this leads to
m!1
c&b,q'@c&b,q'
@b: _b#_r
! "&16'
The following evolution equation for r is assumed:
_r ! hG, G!@c&b,q'
@b: _b when _R40 &17a'
_r ! 0, when _R ! 0 &17b'
where h&0rhr1' denotes a material parameter that determinesthe rate of expansion of the surface gs. With the help of Eq. (17a),m is rewritten by
m!1
c&b,q'&1#h'G &18'
Since the non-IH (strain-hardening stagnation) appears duringreverse deformation after prestrain, the initial value of r may beassumed to be zero, which is different with Yoshida’s assumptionof r with a very small initial value.
There are altogether eight material parameters to be identifiedwhen using Yoshida–Uemori hardening law: Y, B, C, m, b, h, K, andn. Moreover, out of comparison’s sake, a power-law (Swift)hardening was also used and in this case, three parameters areidentified: Y, K and n according to Eq. (11).
3.2. Hill’s 1948 yield stress function
Hill’s 1948 quadratic anisotropic yield function [12] can bewritten as
f&s' ! s !#######################################################################################################################################F&s22#s33'2%G&s33#s11'2%H&s11#s22'2%2Ls223%2Ms231%2Ns212
q
&19'
where s is the equivalent stress, and 1, 2, 3 stand for the RD, TDand ND, respectively. F, G, H, L, M and N are material parameters.
The condition on the initial elastic limit along the RD imposesthe relation G+H!1. In the case of sheet materials, mechanicaltests involving si3 are rather difficult to perform and therefore, inthe following, it is assumed that L and M are kept equal to theirvalue in case of an isotropic behavior, i.e. L!M! 1.5. There arethen three material parameters to identify: F, G and N.
3.3. Bron–Besson yield stress function
The classically used form of [5] yield function is defined by anequivalent stress:
f&s' ! s !X2
k ! 1
ak&sk'a !1=a
&20'
where sk are one order positive and homogeneous functionswhich are convex with respect to s. ak are the weights of eachfunction sk and positive coefficients, the sum of which is equal to1. The functions are defined by
sk ! &ck'1=bk
&21'
S.L. Zang et al. / International Journal of Mechanical Sciences 53 (2011) 338–347 341
where
c1 ! 12
$jS12#S13j
b1 % jS13#S11jb1 % jS11#S12j
b1%
c2 !3b2
2b2 %2
$jS21j
b2 % jS22jb2 % jS23j
b2%
In Eq. (21), Ski ! 1,3 are the principal values of a modified stressdeviator sk, whose components are obtained from the followinglinear transformation of the Cauchy stress s, which can berepresented by a six-component vector (s11,s22,s33,s12,s23,s31)
T:
sk ! Lk : s &22'
where
Lk !
&ck2%ck3'=3 #ck3=3 #ck2=3 0 0 0
#ck3=3 &ck3%ck1'=3 #ck1=3 0 0 0
#ck2=3 #ck1=3 &ck1%ck2'=3 0 0 0
0 0 0 ck4 0 0
0 0 0 0 ck5 0
0 0 0 0 0 ck6
0
BBBBBBBBBB@
1
CCCCCCCCCCA
&23'
The anisotropy of Bron–Besson yield function is represented
by 12 parameters, ck ! 1#2i ! 1#6 , in the form of two fourth order
symmetric tensors L1, L2. The other parameters a, b1, b2 and
a! a1 (a2 ! 1#a1) only influence the shape of the yield surface.Thereby, the yield function has a total of 16 parameters. Note thatthe convexity and derivability of the yield surface are well proven
for aZ1 and bkZ2.As for Hill’s 1948 yield criterion, parameters ck ! 1#2
5 andck ! 1#26 which are related to shear in the sheet thickness are keptequal to 1. There are, therefore, 12 material parameters toidentify: ck ! 1#2
i ! 1#4 , a, b1, b2 and a.
3.4. Large strain framework
These constitutive equations have been implemented withinSiDoLo software, which is a tool box for model developmentbased on differential equations and for material parameteridentification. The general framework is elasto-visco-plasticityin finite strains, by the use of the corotational frame to fulfill thematerial frame indifference requirement. This frame is associatedto the skew-symmetric part W of the velocity gradient L. Let Q c
be the rotation between the current space frame and the corota-tional frame, _Q c tQ c !W . An additive decomposition of the strainrate tensor in the corotational frame is chosen:
_e ! &Q c'TDQ c ! _ee% _e in &24'
with the strain rate tensor D! L#W . Constitutive laws arewritten in the corotational frame, using the corotated Cauchystress tensor s defined by
s! &detF'&Q c'TrQ c &25'
with F the transformation gradient and r the Cauchy stresstensor. The associated derivative is then the Jaumann derivative.Within the SiDoLo environment, homogeneous mechanical testsare reproduced by defining either components of the transforma-tion gradient or the stress tensor.
4. Results and discussion
4.1. Parameter identification
Inverse identification of the material parameters is carried outby optimization using uniaxial tensile, simple shear and balanced
biaxial tensile tests with the software SiDoLo [13]. The costfunction L&A' is defined in the least square sense by Eq. (26)and is minimized with a Levenberg–Marquardt algorithm, start-ing from an initial guess of material parameters A0.
L&A' !XN
n ! 1
Ln&A'
!X
aLTU#Sn &A'%
X
aLTU#Epsn &A'%
X
a,Bausch:LShear#Sn &A'%LBT#S
n &A'
&26'
with N the number of tests in the database. Superscript ‘TU-S’stands for the stress level in tension, ‘TU-Eps’ stands for the widthstrain in tension, ‘Shear-S’ stands for the stress level in shear and‘BT-S- stands for the stress level in balanced biaxial tension. Thesum of a for tension and shear is performed over all orientationsto RD and Bauschinger tests in simple shear are also taken intoaccount (subscript ‘Bausch.’). For each test, the gap betweenexperiments and model is given by
Ln&A' !1Mn
XMn
i ! 1
&Z&A,ti'#Z)&ti''TDn&Z&A,ti'#Z)&ti'' &27'
where Mn is the number of experimental points of the nth test,Z&A,ti'#Z)&ti' the gap between experimental Z) and simulatedoutput variables Z at time ti, and Dn a weighting matrix for thenth test. The experimental database consists of tests with twoobservable variables, namely stress and strain components. Adifferent weighting coefficient is affected for each of theseobservable variables, the value of which is chosen according tothe uncertainty on the experimental measurements. For the shearstress, the value of the weighting coefficient is DsXY ! 3 MPa; forthe uniaxial tensile tests, DsXX ! 5 MPa and DeYY ! 0:002 and forthe balanced biaxial tensile test, DsXX ! 5 MPa. The database ismade up of the uniaxial tensile tests at 01, 451 and 901 to the RDfor the DP500 and 01, 221, 451, 771 and 901 to the RD for the DC04,taking into account the transverse strain, of the monotonic simpleshear tests for the same orientations, of the three cyclic sheartests in the RD and of the balanced biaxial tensile test.
Although the viscous behavior has been observed in manymetallic materials, the rate-independent constitutive models arestill widely used to describe the mechanical behavior of sheets inquasi-static forming processes. The viscous behavior could beneglected by choosing suitable values of viscosity parameters, Kv
and nv, of Eq. (3) in an elasto-visco plastic constitutive model. Inthe present paper, the rate-independence is assumed, then thestrain rate sensitivity parameters were fixed to Kv!5 MPa s1=nv
and nv!4, which leads to a viscous contribution of the order of1.6 MPa at a strain rate of 10#3 s#1. In order to check whether allthe tests were performed within a similar strain rate range, theevolution of the equivalent viscoplastic strain is investigated inthe simulations of uniaxial and balanced biaxial tension and shearin the RD as shown in Fig. 4. In uniaxial tension, the equivalentviscoplastic strain rate for these two steels is almost constant andof the same magnitude except for the sharp increase at the end ofthe test for DP500 since necking occurs. A similar evolution isobserved in simple shear, but relates to different constant values.In addition, these rather constant strain rates in tension andsimple shear leads to a constant overstress. On the contrary, aserious non-linearity of strain evolution with time is observed forthe bulge test, which is caused by the rapid evolution of localiza-tion. From the minimum and maximum strain rates reachedamong these tests (DP500), the largest gap in the viscouscontribution was estimated below 1 MPa which can be neglectedwhen comparing it with the yield stress. Hence, the current fixedstrain rate sensitivity parameters are reasonable to support therate-independent assumption.
S.L. Zang et al. / International Journal of Mechanical Sciences 53 (2011) 338–347342
In general, the initial yield stress in tension and shear andplastic anisotropy coefficients are used to describe the initialanisotropy, and their evolution is related to the hardeningbehavior and strain-induced anisotropy. With the current identi-fication strategy, the accurate prediction of anisotropy and hard-ening behavior is represented by a better description of flowstress in tension and shear and plastic anisotropy coefficients. Itshould be emphasized that in this work, plastic anisotropycoefficients are not directly used in the optimization procedurebut the evolution of the transverse strain with respect to thelongitudinal strain in tension. This data correspond to the rawdata from experiments and r-values are simply calculated from it.In order to check the capability of the current identificationstrategy, three constitutive models are used to predict the
anisotropy and hardening behavior of metallic sheets. The con-stitutive models are based on Hill’s 1948 or Bron–Besson yieldfunction, Swift isotropic hardening model and with or withoutYoshida-Uemori kinematic hardening model, which are termed as‘Hill–Iso’, ‘Hill–Yoshida’ and ‘Bron–Yoshida’, respectively.
Elastic properties were not optimized: Young’s modulus isdirectly measured from tensile tests and values are 191 GPa forDP500 and 176 GPa for DC04. Poisson’s ratio is fixed to 0.29 forboth materials.
4.2. Prediction of the anisotropy and hardening behavior for DP500
Material parameter identification for the three constitutivemodels is performed for the DP500 steel and values are listedin Table 2. With this set of parameters, the strain–stress responsesin the uniaxial and balanced biaxial tension, monotonic andBauschinger simple shear, and transverse strains in tension arepredicted as shown in Figs. 5–7, as well as plastic anisotropycoefficients in Fig. 8.
The experimental and predicted Cauchy stress in uniaxial andbalanced biaxial tensile, and monotonic simple shear tests at 01 tothe RD are shown in Fig. 5(a). Concerning cyclic shear tests, thetransient behavior, strain-hardening stagnation and permanentsoftening are experimentally observed in the Bauschinger simpleshear tests (cf. Fig. 5(b)). Meanwhile, it is found that the strain-hardening stagnation increases with the prestrain. The flow stressof the balanced biaxial tensile test is larger than others, andpresents more non-saturating hardening behavior. Hill–Yoshidaand Bron–Yoshida constitutive models have almost the similarcapability to predict the flow stress, however, the Cauchy stresspredicted by Hill–Iso constitutive model is much lower at the endof the balanced biaxial tension, but higher under reversal simpleshear since its incapability to predict the Bauschinger effect. It canbe seen that Yoshida–Uemori model leads to an overall gooddescription of the stress–strain curve after reloading in theopposite direction. However, the stagnation is too severe, indeed
Table 2
Material parameters for the DP500 and DC04 steel sheets; L and M for Hill’s 1948 yield function and ckiZ5 for Bron–Besson yield function are irrelevant. The material
parameters of Hill–Iso constitutive model are identified from the same material database except for the Bauschinger simple shear tests.
Model Label Unit Hill–Iso Hill–Yoshida Bron–Yoshida
DP500 DC04 DP500 DC04 DP500 DC04
F – 0.488 0.282 0.492 0.289 – –Hill1948 G – 0.461 0.264 0.459 0.267 – –yield function N – 1.589 1.754 1.607 1.807 – –
Fig. 4. Evolution of the equivalent viscoplastic strain p in uniaxial and balancedbiaxial tension and simple shear at 01 to the RD, obtained with SiDoLo software.The material parameters used here correspond to the column termed as ‘Bron–Yoshida’ in Table 2.
S.L. Zang et al. / International Journal of Mechanical Sciences 53 (2011) 338–347 343
as can be seen from experiments in Fig. 5(b), the hardening ratedecreases but remains positive whatever the prestrain. Thesimulated transverse strain eYY in uniaxial tension is comparedin Fig. 6(a). It can be seen that eYY predicted by Hill–Iso orHill–Yoshida constitutive model is similar and exhibits the largestgap with experiments, while eYY is better predicted byBron–Yoshida model.
Fig. 7 shows that the gap between the Cauchy stress simulatedby Hill–Iso or Hill–Yoshida constitutive model and experimentsincreases with plastic deformation in uniaxial tension at 451 tothe RD, while the flow stress predicted by Bron–Yoshida fits wellwith experiments. In addition, the Cauchy stress calculated byHill–Iso or Hill–Yoshida is lower than the experimental value. Asfor the simulated shear stress for these three models, thedifferences are rather small. The prediction of the transversestrain eYY in uniaxial tension at 451 to the RD is similar to thatof tension in the RD as illustrated in Fig. 6(b). Similar results havebeen obtained at 901 to the RD and are not displayed here.
4.3. Prediction of the anisotropy and hardening behavior for DC04
In order to check the identification strategy for the moreanisotropic material, the prediction of the anisotropy and hard-ening is also performed for the DC04 steel. Similarly to the DP500
steel, using material parameters in Table 2, the strain–stressresponses in uniaxial and balanced biaxial tension, monotonicand Bauschinger simple shear, and transverse strains in tensionare predicted as shown in Figs. 9–11, and plastic anisotropycoefficients are shown in Fig. 12.
The experimental and predicted Cauchy stress in uniaxial andbalanced biaxial tensile, and monotonic and Bauschinger simpleshear tests in the RD is shown in Fig. 9. The transient behavior,hardening stagnation and permanent softening are experimen-tally observed. Conversely to the DP500 steel, the flow stress ofthe balanced biaxial tensile test is much larger than that ofuniaxial tension or simple shear. Similarly to the DP500 steel,
0
400
800
0 0.1 0.2 0.3 0.4 0.5 0.6
! (M
Pa)
"/#
Biaxial tension
Tension
Shear
ExpBron-Yoshida
Hill-IsoHill-Yoshida
-400
0
400
-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4
! (M
Pa)
#
exp.Bron-Yoshida
Hill-Iso
Fig. 5. Predicted Cauchy stress in uniaxial and balanced biaxial tensile tests andsimple shear tests at 01 to the RD (DP500). e stands for the logarithmic straincomponent eXX in tension and balanced biaxial tension, where ~X is parallel to therolling direction. (a) Monotonic tests. (b) Cyclic shear tests.
-0.1
-0.05
0
0 0.05 0.1 0.15 0.2
" YY
"XX
ExpBron-Yoshida
Hill-IsoHill-Yoshida
-0.1
-0.05
0
0 0.05 0.1 0.15 0.2" Y
Y
"XX
ExpBron-Yoshida
Hill-IsoHill-Yoshida
Fig. 6. Evolution of transverse strain with longitudinal strain in uniaxial tension(DP500). (a) 01/RD. (b) 451/RD.
0
200
400
600
0 0.1 0.2 0.3 0.4 0.5 0.6
! (M
Pa)
"/#
Tension
Shear
ExpBron-Yoshida
Hill-IsoHill-Yoshida
Fig. 7. Predicted Cauchy stress in uniaxial tensile tests and simple shear tests at451 to the RD (DP500).
S.L. Zang et al. / International Journal of Mechanical Sciences 53 (2011) 338–347344
the Cauchy stress predicted by Hill–Iso constitutive model isslightly lower than others at the end of balanced biaxial tension.The transverse strain eYY predicted by Hill–Iso or Hill–Yoshidaconstitutive model is the same and also exhibits the largest gapwith experiments, while eYY predicted by Bron–Yoshida model ismuch closer to the experiments, as compared in Fig. 10(a).
Fig. 11 shows that these three constitutive models have almostthe similar capability to predict flow stress in uniaxial tension
and simple shear at 451 to the RD. However, the flow stresspredicted by Bron–Yoshida model fits better with experiments.Still, it can be noted that all these constitutive models cannot wellpredict the initial yield stress, i.e. the current identificationstrategy loses some accuracy on the description of the initialyield for the more anisotropic material. As for the prediction oftransverse strain eYY in uniaxial tension, a good description isobtained with Bron–Yoshida constitutive model as shownin Fig. 10(b). The prediction of the stress and transverse straineYY in uniaxial tension at the other directions to the RD is similarto those at 01 and 451.
0.9
1
1.1
1.2
0 15 30 45 60 75 90
r $
$ (°)
ExpBron-Yoshida
Hill-IsoHill-Yoshida
Fig. 8. Prediction of plastic anisotropy coefficients (DP500).
0
200
400
600
800
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
! (M
Pa)
"/#
Biaxial tension
Tension
Shear
ExpBron-Yoshida
Hill-IsoHill-Swift
-200
-100
0
100
200
-0.4 -0.2 0 0.2 0.4#
ExpBron-Yoshida
Hill-Iso
! (M
Pa)
Fig. 9. Predicted Cauchy stress in uniaxial and balanced biaxial tensile tests andsimple shear tests at 01 to the RD (DC04). e stands for the logarithmic straincomponent eXX in tension and balanced biaxial tension, where ~X is parallel to therolling direction. (a) Monotonic tests. (b) Cyclic shear tests.
-0.15
-0.1
-0.05
0
0 0.05 0.1 0.15 0.2 0.25
" YY
"XX
ExpBron-Yoshida
Hill-IsoHill-Swift
-0.15
-0.1
-0.05
0
0 0.05 0.1 0.15 0.2" Y
Y"XX
ExpBron-Yoshida
Hill-IsoHill-Swift
Fig. 10. Evolution of transverse strain with longitudinal strain in uniaxial tension(DC04). (a) 01/RD. (b) 451/RD.
0
100
200
300
400
0 0.2 0.4 0.6
! (M
Pa)
"/#
Tension
Shear
ExpBron-Yoshida
Hill-IsoHill-Yoshida
60
75
90
0.005 0.01
120
160
0 0.01 0.02
Fig. 11. Predicted Cauchy stress in uniaxial tensile tests and simple shear tests at451 to the RD (DC04).
S.L. Zang et al. / International Journal of Mechanical Sciences 53 (2011) 338–347 345
4.4. Discussion
For both materials, Hill’s 1948 yield function has a poorcapability to model transverse strain since it lacks degrees offreedom and since the weight was put on stress level, in order tocorrectly predict the Bauschinger effect. Indeed, from Eq. (19), itcomes that only three material parameters are independent in theplane stress condition. While seven stress states and threetransverse strains for the DP500 steel, and 11 stress states andfive transverse strains for the DC04 steel, are used to identify thematerial parameters of the constitutive models. Thus, seriousover-constraints should exist in the determination of the materialparameters of Hill’s 1948 yield function. To achieve the overallcompromise of the prediction over all the tests, a poor descriptionis reached for each test.
On the contrary, Bron–Besson yield function, with 12 inde-pendent parameters in case of plane stress state is expected toaccurately predict both the anisotropy and hardening behavior ofmetallic sheets. However, one large error point is observed in theprediction of the anisotropy coefficient for the DP500 steel asillustrated in Fig. 8, although Bron–Yoshida constitutive modeldescribes accurately the flow stress. This is mainly caused by themeasure of the plastic anisotropy coefficient. Usually, this coeffi-cient is measured in a given strain range. With this method, theexperimental plastic anisotropy coefficient is significantly influ-enced by the choice of the strain range. In this condition, theexperimental coefficient cannot correctly describes the transversestrain in tension. In addition, the distribution of experimentalpoints of eYY for the DC04 steel is uniform with a constantsampling frequency, which is different with that of the DP500steel. It indicates that the measured anisotropy coefficient for theDC04 steel can accurately express the transverse strain eYY sinceno necking occurs, as illustrated in Fig. 12.
Considering results obtained both with Hill48–Iso andHill48–Yoshida models, it can be seen that the predicted plasticanisotropy coefficients are similar, though rather far from experi-mental results. This similarity comes from the fact that theevolution law of X, which brings the largest contribution to thekinematic part of hardening, is written according to [14].
The main point for using Yoshida–Uemori model is to predictthe work-hardening stagnation recorded after a Bauschingertest [15]. This stagnation has been evidenced both with tension-compression tests [16] and cyclic simple shear tests. This laststrain path is particularly well suited for strain path reversal, inthe sense that it allows an equivalent plastic prestrain up to 0.2 tobe investigated and to reach after straining an equivalent plastic
strain of 0.5. In order to highlight this stagnation, the work-hardening rate ds=dg has been calculated by fitting splines on thestress–strain curves of Fig. 9(b), considering both the experimentsand the simulation, and by finite difference derivation. Fig. 13shows that the experimental work-hardening rate remainsstrictly positive whatever the prestrain value, which is consistentwith previously published results [15]. Therefore, though themagnitude of the plateau is well represented, the hardening ratepredicted by Yoshida–Uemori is too small compared to theexperiments. The prediction of the mechanical behavior inreloading after a prestrain could be further improved by modify-ing the parameter C, as suggested in [17].
The current identification strategy focuses simultaneously onboth the initial and strain-induced anisotropy, thus avoiding toget the initial yield stress and plastic anisotropy coefficients.Hence, better prediction on transverse strain in tension and flowstress in uniaxial and balanced biaxial tension and simple shear isachieved.
Among the three constitutive models, Bron–Yoshida constitu-tive model accurately predicts both the flow stress and transversestrain eYY for all the tests as expected, although the currentidentification strategy loses some accuracy in the prediction ofthe initial yield stress. The comparison of the final values of thecost function is shown in Fig. 14. The cost function is an indicatorof the gap between experimental and simulated values. Itdepends also on the number of tests in the experimental databaseand on the weighting coefficients. In this work, the experimentaldatabase is the same for all simulations and the weightingcoefficients are kept constant. Comparing with Hill–Yoshidaconstitutive model, the reduction of the cost function withBron–Yoshida model is very clear, it is around 7% for DP500 steeland 45% for DC04 steel.
1.5
1.9
2.3
2.7
0 15 30 45 60 75 90
r $
$ (°)
ExpBron-Yoshida
Hill-IsoHill-Yoshida
Fig. 12. Prediction of plastic anisotropy coefficients (DC04).
0
250
500
750
1000
-0.3 -0.2 -0.1 0 0.1 0.2 0.3
d%/d
# (M
Pa)
#
ExpBron-Yoshida
0
250
500
750
1000
-0.3 -0.2 -0.1 0 0.1 0.2 0.3#
ExpBron-Yoshida
d%/d
# (M
Pa)
Fig. 13. Work-hardening rate, calculated as dt=dg evolution versus shear strainduring cyclic shear tests. Only the reversed loading path is plotted. (a) DP500.(b) DC04.
S.L. Zang et al. / International Journal of Mechanical Sciences 53 (2011) 338–347346
5. Conclusion
In this paper, characterization of the anisotropy and hardeningbehavior of DP500 and DC04 steels is presented as well as theirmodeling within an elasto-viscoplasticity framework, based onBron–Besson yield function [5] and Yoshida–Uemori kinematichardening model [6]. Tests in uniaxial tension, monotonic andBauschinger simple shear and balanced biaxial tension are con-sidered. The anisotropy and hardening behavior are identifiedfrom the stress–strain curves and transverse strain in tension, toconsider the subsequent evolution of the plasticity surface afterthe initial yield. It is shown that this identification strategy losessome accuracy on the description of the initial yield stresses, butgains more accuracy in the prediction of the global hardening
behavior. Hill’s 1948 yield function is also chosen for compar-ison’s sake. The results show that Bron–Besson yield function canwell describe the anisotropy of the yield stress and transversestrain in uniaxial tension at the same time, whereas Hill’s 1948yield function only works for one of them. It indicates that suchadvanced anisotropic yield function should be used in theprediction of the anisotropy when more than tension is expectedto be accurately reproduced, even for the steel with a nearlyisotropic mechanical behavior.
Acknowledgments
The authors would like to thank the Region Bretagne andDirection Generale des Entreprises for their financial support.
References
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150
200
250
300
350
Hill-Yoshida Bron-Yoshida
Fina
l val
ue o
f cos
t fun
ctio
n
Constitutive model
DP500DC04
Fig. 14. Comparison of the final values of cost function with different constitutivemodels.
S.L. Zang et al. / International Journal of Mechanical Sciences 53 (2011) 338–347 347
