A new strategy for the simultaneous identification of constitutive laws parameters of metal sheets using a single test

Abstract

An inverse analysis methodology for determining the parameters of plastic constitutive models is proposed. This involves the identification of the yield criterion and work-hardening law parameters, which best describe the results of the biaxial tensile test on cruciform samples of metal sheets. The influence and sensitivity of the constitutive parameters on the biaxial tensile test results is studied following a forward analysis, based on finite element simulations. Afterwards, the inverse analysis methodology is established, by evaluating the relative difference between numerical and experimental results of the biaxial tensile test, namely the load evolution in function of the displacements of the grips and the equivalent strain distribution, at a given moment of the test, along the axes of the sample. This methodology is compared with a classical identification strategy and proves to be an efficient alternative, allowing to avoid time-consuming tests, some of them hard to analyse and liable to uncertainties.

Full text

A new strategy for the simultaneous identification of constitutive laws
parameters of metal sheets using a single test
P.A. Prates
⇑
, M.C. Oliveira, J.V. Fernandes
CEMUC, Department of Mechanical Engineering, University of Coimbra, Rua Luís Reis Santos, Pinhal de Marrocos, 3030-788 Coimbra, Portugal
article info
Article history:
Received 5 July 2013
Received in revised form 10 December 2013
Accepted 19 December 2013
Keywords:
Constitutive law parameters
Inverse analysis
Cruciform biaxial tensile test
Finite element method
abstract
An inverse analysis methodology for determining the parameters of plastic constitutive models is
proposed. This involves the identification of the yield criterion and work-hardening law parameters,
which best describe the results of the biaxial tensile test on cruciform samples of metal sheets. The
influence and sensitivity of the constitutive parameters on the biaxial tensile test results is studied
following a forward analysis, based on finite element simulations. Afterwards, the inverse analysis meth-
odology is established, by evaluating the relative difference between numerical and experimental results
of the biaxial tensile test, namely the load evolution in function of the displacements of the grips and the
equivalent strain distribution, at a given moment of the test, along the axes of the sample. This method-
ology is compared with a classical identification strategy and proves to be an efficient alternative,
allowing to avoid time-consuming tests, some of them hard to analyse and liable to uncertainties.
Ó2013 Elsevier B.V. All rights reserved.
1. Introduction
The accuracy of the numerical simulation results in sheet metal
forming depends on the selected constitutive model and the
strategy used for the parameters identification [1–4]. Several
phenomenological yield criteria (e.g. [5–10]) and hardening laws
(e.g. [11–16]) have been proposed in order to improve the descrip-
tion of the plastic behaviour of metal sheets. Increasing the flexibil-
ity of the constitutive models is often associated with a larger
number of parameters to identify. This requires a wide set of
experimental tests and complex identification strategies. The con-
stitutive parameters are usually identified from linear strain path
tests (namely tensile, bulge and shear tests) with homogeneous
deformation in the measuring region, using classical methodolo-
gies. As the rolling process makes the metal sheets anisotropic, dif-
ferent mechanical behaviours are expected for different loading
directions and conditions. However, sheet metal forming processes
are carried out with inhomogeneous deformation and under mul-
tiaxial strain paths. Therefore, limiting the characterization of the
mechanical behaviour of metal sheets to a restricted number of
tests with linear strain paths and homogeneous deformation can
lead to a somewhat incomplete characterization of the overall
plastic behaviour of the material [17].
From mechanical tests with heterogeneous strain fields it is
possible to obtain a larger amount of information than the one
found in case of tests with homogeneous strain fields. Therefore,
heterogeneous strain fields can more suitably describe the influ-
ence of the strain path on the plastic behaviour of metals than
homogeneous strain fields [18]. Material parameters obtained
from homogeneous strain path tests are more appropriate for
describing the material behaviour for one particular strain path,
but can be unsuitable for other strain paths. To overcome this
problem, it is necessary to develop tests allowing heterogeneous
stress and strain fields and, eventually, strain path changes. The
material parameters obtained through these tests will describe
the overall mechanical behaviour of the material, taking into ac-
count the mutual influence of the various strain paths occurring
in the sample, even if they are not fully appropriate for describing
each particular strain path [18–20]. The material parameters ob-
tained through such tests will be also suitable for describing the
plastic behaviour of metal sheets during complex forming opera-
tions, in view of the heterogeneous nature of the deformation.
The increasing development of optical full-field measurement
techniques for analysing heterogeneous strain fields, such as the
digital image correlation (DIC) technique, has led to the develop-
ment of new tests and methodologies for characterising the plastic
behaviour of materials [21]. One possible approach consists on
using inverse analysis methodologies, which are based on the
determination of the material parameters that minimise the gap
between numerically predicted and experimental test results
[22]. These methodologies have been recently explored in the liter-
ature for the parameters identification of constitutive laws, by
combining DIC measurements on the test samples with numerical
simulation results of the test [18–20,23–25]. In this context, sev-
eral works in literature, which propose the coupling of optical
0927-0256/$ - see front matter Ó2013 Elsevier B.V. All rights reserved.
http://dx.doi.org/10.1016/j.commatsci.2013.12.043
⇑
Corresponding author. Tel.: +351 239790700; fax: +351 239790701.
E-mail address: pedro.prates@dem.uc.pt (P.A. Prates).
Computational Materials Science 85 (2014) 102–120
Contents lists available at ScienceDirect
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journal homepage: www.elsevier.com/locate/commatsci

measurement results and inverse analysis methodologies together
with numerical simulations results for the identification of consti-
tutive laws parameters, are highlighted in the following.
Güner et al. [23] proposed an inverse analysis procedure for the
identification of the Yld2000-2d yield criterion parameters corre-
sponding to the initial yield locus of representative materials. This
study, strictly numeric, uses a notched specimen submitted to a
uniaxial tensile test, enabling strain paths near uniaxial tension.
The required data for the inverse identification of the yield crite-
rion parameters are variables such as the major and minor princi-
pal strains in the sheet plane, the tool force, and the equibiaxial
yield stress value (which is assumed known a priori). The objective
function is a combination of principal strain, tool force (at selected
tool displacements) and equibiaxial yield stress differences
between numerically generated and experimental reference val-
ues, and is minimised using the Levenberg–Marquardt algorithm.
Different alternative orientations of the specimen with the rolling
direction (0°,45°,90°) and configurations of the objective function
(setting the strain, or tool force, components to zero) were consid-
ered to test the inverse procedure. The authors highlight the
importance of including strain information on the objective func-
tion, which leads to an improvement on its minimisation.
Pottier et al. [18] developed a testing procedure based on the
out-of-plane deformation of a sample, using stereo image correla-
tion, for the simultaneous identification of the constitutive param-
eters of Hill’48 yield criterion and Ludwick work-hardening law of
a pure titanium sheet. The identification procedure consists of a
finite element update inverse method and the parameters are
determined using Levenberg–Marquardt minimisation strategy,
where the gap between experimental and finite element simula-
tion results of the surface displacement fields and the global force
is minimised. The authors highlight the importance of increasing
the strain field heterogeneity for a better assessment of the mate-
rial behaviour.
In another work, an inverse analysis methodology based on a
least-squares formulation along with Gauss–Newton minimisation
strategy was developed in order to simultaneously determine the
constitutive parameters for Hill’48 yield criterion and Swift
work-hardening law of a stainless steel [20]. In this case, the
parameter identification is performed from the results of three dif-
ferent complex tests, all of them comprising heterogeneous strain
fields: a uniaxial tensile test on a perforated tensile specimen, a
uniaxial tensile test on a complex shaped specimen and a biaxial
tensile test of a perforated cruciform specimen. Furthermore, the
sets of parameters obtained from each test are applied to simulate
the three complex tests previously described. This allowed
concluding that a good practice is to develop the mechanical test
in accordance with the sheet metal forming process in study
[20]. This methodology was also adopted for performing the iden-
tification of Hill’48 yield criterion and Swift work-hardening law
parameters of a mild steel, from the results of a biaxial tensile test
of a perforated cruciform specimen [19]. This strategy allowed the
determination of averaged parameters of the yield criterion and
hardening law, which are better suited for the simulation of real
sheet metal forming processes than the ones obtained from classi-
cal identification strategies [19].
The idea of testing cruciform specimens dates back to the 1960s
[26]. Such tests show potential for application in characterising the
plastic behaviour of materials, i.e. for estimating the parameters of
the anisotropic yield criterion and the work-hardening law,
namely: (i) strain paths ranging from uniaxial tension (in the arms
region of the specimen) to biaxial tension (in the central region of
the specimen), (ii) high strain gradients, from the central region of
the specimen to the extremity of the arms and (iii) no sliding con-
tact occurs with tools, avoiding friction. Also, by changing the load
and/or the displacement ratio between the two perpendicular
loading axes, it is possible to obtain different biaxial strain and
stress states, in the central region of the specimen [27]. However,
this test allows only attaining low values of equivalent plastic
strain (close to those obtained in uniaxial tension) before instabil-
ity occurs and no occurrence of out-of-plane shear stress is ob-
served (which prevents the determination of the constitutive
parameters associated with out-of-plane stress components, as
usually occurs when using classical methodologies). The aim of this
work consists in developing and evaluating the performance of an
inverse analysis methodology for the identification of the plastic
constitutive parameters (anisotropic yield criterion and work-
hardening law), which describe the plastic behaviour of metal
sheets, from a single biaxial tensile test of a cruciform specimen.
The current approach aims to be simple, from an experimental
point of view, and for this purpose one just analyses the load evo-
lution during the test and the equivalent strain distribution along
the axes of the specimen, at a given moment of deformation, as
an alternative to follow the strain fields on the specimen surface
during the test, as previously performed by other authors [19].
2. Numerical model
The geometry of the cruciform specimen was studied using
finite element method results in order to reproduce, as far as
possible, inhomogeneous deformation with the occurrence of
strain paths that are commonly observed in sheet metal forming
processes [28]. An overview of the optimisation procedure for
the sample geometry is presented in Appendix A.Fig. 1 shows
the selected geometry and the relevant dimensions of the cruci-
form specimen in the sheet plane. The 0xand the 0yaxes coincide
with the rolling direction (RD) and the transverse direction (TD) of
the sheet, respectively. The cruciform specimen is submitted to
equal displacements in both 0xand 0ydirections, applied by the
grips, as indicated in Fig. 1. The displacements along the 0xand
0yaxes are measured at points A and B, respectively. The sheet
thickness considered in this study is 1.0 mm.
The material is considered orthotropic. Due to geometrical and
material symmetries, only one eight of the specimen was consid-
ered in the numerical simulation model. The specimen was discre-
tised with tri-linear 8-node hexahedral solid elements with an
average in-plane size of 0.5 mm and two layers through-thickness.
Numerical simulations were carried out with DD3IMP in-house
Fig. 1. Geometry and dimensions of the cruciform specimen. The grips, represented
in grey, hold the specimen by grabbing it along the dashed grey lines. A and B
represent the points for measuring the displacements,
D
l.
P.A. Prates et al. / Computational Materials Science 85 (2014) 102–120 103

finite element code, developed and optimised to simulate sheet
metal forming processes [29].
The constitutive model adopted in developing of the proposed
methodology considers Hill’48 yield criterion [30] with the associ-
ated flow rule and Swift work-hardening law [31]. The Hill’48 yield
criterion describes the yield surface for the case of an orthotropic
material as follows:
Fð
r
yy

r
zz
Þ
2
þGð
r
zz

r
xx
Þ
2
þHð
r
xx

r
yy
Þ
2
þ2L
s
2
yz
þ2M
s
2
xz
þ2N
s
2
xy
¼Y
2
ð1Þ
where
r
xx
,
r
yy
,
r
zz
,
s
xy
,
s
xz
and
s
yz
are the components of the
effective Cauchy stress tensor (
r
) defined in the system of axes of
orthotropy of the metal sheet; F,G,H,L,Mand Nare the anisotropy
parameters to be identified and Yis the yield stress, which evolu-
tion is defined by the work-hardening law. In this study, it is
assumed the condition G+H= 1 and so the work-hardening law is
represented by the uniaxial tensile stress along the RD. The work-
hardening is described by Swift law as follows:
Y¼Cð
e
0
þ
e
p
Þ
n
ð2Þ
where 
e
p
is the equivalent plastic strain and C,
e
0
and nare the
material parameters. The yield stress can also be written as a func-
tion of C,nand Y
0
, where Y
0
=C(
e
0
)
n
. The elastic behaviour is consid-
ered isotropic and is described by the generalised Hooke’s law.
3. Identification strategy
The proposed identification strategy consists on the simulta-
neous determination of the constitutive parameters of Hill’48 crite-
rion and Swift law, using results of the biaxial tensile test on the
cruciform sample. The first step of this work consists on a forward
analysis using finite element simulation results of this test, in order
to study the sensitivity of the results to variations of the values of
the constitutive models parameters, revealed by the evolution and/
or distribution of variables such as the ones shown in Fig. 2. This
forward study allowed the developing of an inverse analysis meth-
odology, for identifying the constitutive laws parameters.
3.1. Forward analysis
In this forward study, the analysis is focused on:
(i) The evolutions of the load, P, with the specimen boundaries
displacement,
D
l, during the test, for the axes 0xand 0y;
D
l
is measured for points A and B in Fig. 1.
(ii) The distributions of the equivalent strain, 
e
v
M
, along the axes
of the sample (i.e. 
e
v
M
as a function of the distance to the
centre of the sample, d), for a given boundaries displace-
ment,
D
l, preceding and close to the displacement at the
maximum load; the equivalent strain is determined using
von Mises definition:

e
v
M
¼2ð
e
2
1
þ
e
2
2
þ
e
1
e
2
Þ=3

1=2
ð3Þ
where
e
1
and
e
2
are respectively the major and the minor
principal strains, in the sheet plane. The principal strain axes
are parallel to the axes of the specimen (in case of the 0xaxis,
e
1
is equal to
e
xx
and
e
2
is equal to
e
yy
, and in case of the 0y
axis,
e
1
is equal to
e
yy
and
e
2
is equal to
e
xx
– see Fig. 1).
(iii) The distributions of the strain path ratio, defined by
e
2
/
e
1
,
along the axes 0xand 0y, for the same boundaries displace-
ment,
D
l, as previously stated in (ii); observations performed
during the test, at different values of
D
l, showed that the
strain paths are quite linear.
It is worth noting that the strain variables,
e
1
and
e
2
, can be
experimentally measured, using DIC technique [32], or even the
classical Circle Grid Strain analysis [33], which allows establishing
a correspondence between the numerical and experimental results
for developing the inverse analysis strategy.
Table 1 summarises the mechanical properties of the illustrative
cases of materials used in the forward study. A material with isotro-
pic behaviour, described by von Mises yield criterion, is denoted as
‘‘reference’’. The remaining materials diverge from the reference
one due to the value of one parameter that was increased by 50%,
compared to the reference one. The following designations are
adopted: ‘‘Y0_300’’ and ‘‘n_0.300’’, for materials presenting also iso-
tropic behaviour, but with different yield stress and work-harden-
ing exponent values, respectively; ‘‘F_0.75’’, ‘‘H_0.75’’ and
‘‘N_2.25’’, for materials with the same hardening behaviour as the
‘‘reference’’ material, but with different Hill’48 anisotropy parame-
ters. The condition G+H= 1 is assumed for all cases; consequently,
in case of the material ‘‘H_0.75’’, the Gvalue is equal to 0.25. Also,
this condition implies that Y
0
=
r
0
and (F+H)
1/2
=
r
0
/
r
90
where
r
0
and
r
90
are the tensile stresses along the rolling and transverse
directions, respectively. In all simulations, the Land Manisotropic
parameters are kept equal to 1.5, since the results of the biaxial cru-
ciform test are not sensitive to these parameters.
e
0
is assumed
fixed with a small value as for most materials not subjected to
pre-strain [12], in this case of 0.005. The elastic behaviour considers
a Young modulus (E) equal to 210 GPa and a Poisson ratio (
m
)
equal to 0.3.
The forward analysis is performed by comparing the numerical
simulation results obtained for the last five materials presented in
Table 1 with the results of the ‘‘reference’’ material (see Fig. 2), in
terms of relative difference. In this context, d
P
is defined as the
relative difference between the load for a given material, P,
and the load for the reference one, P
ref
, at a certain
D
lvalue, i.e.
d
P
=(PP
ref
)/P
ref
;d

e
v
M
is the relative difference between von Mises
equivalent strain of a given material, 
e
v
M
, and the reference one,

e
v
M
ref
, at a given dvalue, i.e. d

e
v
M
¼ð

e
v
M

e
v
M
ref
Þ=
e
v
M
ref
. The strain
paths can also be analysed in terms of the relative difference of
the ratio
e
2
/
e
1
, at a given dvalue. However, a more intuitive param-
eter is adopted to define the dissimilarity of the strain paths,
cos(
u
), which is the cosine of the angle
u
that corresponds to
the angular difference between the two vectors representing the
strain tensors, such that:
cosð
u
Þ¼ð
e

e
ref
Þ=ðk
e
kk
e
ref
kÞ ð4Þ
where
e
and
e
ref
are the vectors representing the strain tensors, for a
given material and the ‘‘reference’’ case, respectively; ||
e
|| and ||
e
ref
||
are the norm of both vectors. In this context, the similarity between
strain paths for a given material and for the ‘‘reference’’ case is full
when cos(
u
) = 1 and the strain paths deviate from each other when
cos(
u
) moves away from 1. In order to correctly calculate the rela-
tive differences for a certain value of dand
D
l, the numerical and
reference variables, shown in Fig. 2, were obtained for the same va-
lue of dand
D
l. This is achieved by performing linear interpolations
of the results.
Fig. 2 shows the effects of an increase of 50% in each parameter
(Y
0
,n,F,Hand N), relatively to the reference material. The results
presented are Pvs.
D
land d
P
vs.
D
l(Fig. 2(a and b), respectively),

e
v
M
vs. dand d

e
v
M
vs. d(Fig. 2(c and d), respectively) and
e
2
/
e
1
vs.
dand cos(
u
) vs. d(Fig. 2(e and f), respectively). The results in
Fig. 2 concern both the 0xand 0yaxes. The results in Fig. 2(c–f)
are plotted for
D
l= 4 mm, equal for both 0xand 0yaxes. This
results concern the distance, d, from the centre of the cruciform
specimen (d= 0 mm) up to a distance corresponding to the mini-
mum value of
e
2
/
e
1
(see Fig. 2(e), where this minimum occurs for
advalue slightly less than 40 mm; after such dvalue,
e
2
/
e
1
104 P.A. Prates et al. / Computational Materials Science 85 (2014) 102–120

(b)(a)
(d)(c)
(f) (e)
reference Y0_300 n_0.300 F_0.75_0x
F_0.75_0y H_0.75_0x H_0.75_0y N_2.25
0
4
8
12
16
20
[kN]
P
[mm]lΔ
-20
-10
0
10
20
30
40
50
60
[mm]lΔ
[%]
P
δ
0.00
0.05
0.10
0.15
0.20
0.25
[mm]d
vM
ε
-80
-60
-40
-20
0
20
40
60
[mm]d
vM
[%]
ε
δ
-0.80
-0.40
0.00
0.40
0.80
1.20
1.60
[mm]d
21
εε
0.95
0.96
0.97
0.98
0.99
1.00
1.01
01234 0 1 2 3 4
010203040 010203040
0 10203040 0 10203040
[mm]d
()
cos
ϕ
Fig. 2. Numerical simulation results for the materials presented in Table 1: (a) Pvs.
D
l; (c) 
e
v
M
vs. d; (e)
e
2
=
e
1
vs. d. Relative differences with respect to the ‘‘reference’’
material: (b) d
P
vs.
D
l; (d) d

e
v
M
vs. dand (f) cos(
u
) vs. ddistributions. The results are shown for both 0xand 0yaxes (in the legend, the absence of a label, 0xor 0y, means that
the results are equal for both axes).
Table 1
Constitutive parameters of the materials used in the forward analysis.
Material Hill’48 Anisotropy Parameters Swift Law Parameters (F+H)
1/2
(=
r
0
/
r
90
)
FGHNY
0
[MPa] C[MPa] n
Reference 0.5000 0.5000 0.5000 1.5000 200.00 577.08 0.200 1.000
Y0_300 0.5000 0.5000 0.5000 1.5000 300.00 865.62 0.200 1.000
n_0.300 0.5000 0.5000 0.5000 1.5000 200.00 980.25 0.300 1.000
F_0.75 0.7500 0.5000 0.5000 1.5000 200.00 577.08 0.200 1.100
H_0.75 0.5000 0.2500 0.7500 1.5000 200.00 577.08 0.200 1.100
N_2.25 0.5000 0.5000 0.5000 2.2500 200.00 577.08 0.200 1.000
P.A. Prates et al. / Computational Materials Science 85 (2014) 102–120 105

increases approaching zero – not shown in the figure). The choice
of this range for the dvalues intends to avoid: (i) considering two
points with the same strain path (the strain paths that occur for d
values between about 20 and 40 mm are repeated between the
latter dvalue and the end of the arms of the specimen) and (ii)
measuring the variables 
e
v
M
and cos(
u
) close to the heads of the
sample, where the comparison between experimental and numer-
ical results can be influenced by the boundary conditions, if the
experimental boundary conditions are not properly reproduced
numerically. In the following, the results will be always considered
within this range of dvalues.
The results shown in Fig. 2 enable performing the study of the
trends required to develop the inverse analysis strategy for the
identification of the constitutive parameters. The influence of an
increase of 50% on the value of each of the parameters Y
0
,n,F,H
and Nin the results of the cruciform test can be summarised as
follows:
(i) The increase of Y
0
increases the level of the Pvs.
D
lcurves
(Fig. 2(a)), so that the corresponding d
P
value is constant dur-
ing the test (Fig. 2(b)). However, the distribution of the
equivalent strain (Fig. 2(c)) and the strain paths (Fig. 2(e)),
along the axes of the sample, are not changed (see also
Fig. 2(d and f)).
(ii) The increase of the work-hardening exponent, n, increases
the slope of the Pvs.
D
lcurves (Fig. 2(a)), without changing
the Pvalue at the beginning of the plastic deformation. Con-
sequently, the d
P
value increases during the test (see
Fig. 2(b)). The distributions of the equivalent strain, along
the axes of the sample, show noticeably complex changes
(Fig. 2(c)): the d

e
v
M
value is positive for dbetween 0 and
18 mm and negative for the remaining region of the arms
(see also Fig. 2(d)). The strain paths, along the axes of the
sample, are almost coincident with the reference material
results (see Fig. 2(e and f)).
(iii) The increase of Fand H, separately, leads to dissimilarities
between 0xand 0yresults. Concerning the load evolution
during the test, the increase of Fleads to an imperceptible
decrease of the Pvs.
D
levolution for the 0xaxis and to a
decrease of the level of the Pvs.
D
lcurve for the 0yaxis
(see Fig. 2(a and b)). The increase of Hleads to an increase
of the Pvs.
D
levolution for the 0xaxis and to a decrease
of the level of the Pvs.
D
lcurve for the 0yaxis (see
Fig. 2(a and b)). Moreover, it is worth noting that the dissim-
ilarity between the 0xand 0yaxes for the Pvs.
D
lresults is
nearly the same whatever the parameter increased, For H
(see Fig. 2(b)). This is certainly related with the fact that
the value of (F+H)
1/2
, which represents
r
0
/
r
90
, is equal to
1.1 for both materials ‘‘F_0.75’’ and ‘‘H_0.75’’ (see Table 1).
Concerning the equivalent strain distribution along the axes
of the specimen, the increase of Fleads to positive values of
d

e
v
M
in the central region of the cruciform specimen (d
between 0 and at about 25 mm) and negative values of
d

e
v
M
in the remaining region of the arms (see Fig. 2(c and
d)). The increase of Hleads to the opposite effect of the
increase of F(i. e. negative values of d

e
v
M
in the central region
of the cruciform specimen (dbetween 0 and at about
25 mm) and positive values of d

e
v
M
in the remaining region
of the arms (see Fig. 2(c) and d)). Concerning the strain path
distribution along the axes of the specimen, the increase of F
leads to small changes in the ratio
e
2
/
e
1
and in the cos(
u
)
values, along both 0xand 0yaxes (see Fig. 2(e and f)). The
increase of Hleads to noticeable changes in the ratio
e
2
/
e
1
,
also clearly perceptible in terms of cos(
u
), for both 0xand
0yaxes (see Fig. 2(e and f)).
(iv) The increase of Nleads to an imperceptible decrease of the P
vs.
D
lresults along both axes (see Fig. 2(a and b)). The 
e
v
M
vs.
dresults show noticeable complex changes with N
(Fig. 2(c)): the d

e
v
M
value is positive at dbetween 0 and
15 mm and negative for the remaining region of the arms.
The strain paths, along the axes of the sample, are almost
coincident with the reference material results (see Fig. 2(e
and f)).
In summary, it can be concluded from Fig. 2(a and b) that the
load evolution during the test is mainly influenced by variations
in the values of the work-hardening law parameters, Y
0
and n,
and in the value of (F+H)
1/2
. The equivalent strain distribution is
not influenced by changes in the Y
0
parameter. Variations in the
values of the remaining parameters (n,F,Hand N) present a com-
plex effect on the equivalent strain distributions. Fig. 2(f) shows
that the cos(
u
) value is close to 1 in the entire measurement
region, not being enough sensitive to variations in the parameters,
except the Hparameter. Thus, the dissimilarity of the strain paths
is not suitable for estimating the constitutive parameters. In this
context, the d
P
vs.
D
lresults will be used for estimating the
work-hardening law parameters and the (F+H)
1/2
value, while
the d

e
v
M
vs. dresults will be only used for estimating Hill’48 param-
eters. The dissimilarity of the strain paths (
e
2
/
e
1
and cos(
u
)) will
not be used in the optimisation procedure.
3.2. Inverse analysis algorithm
The forward analysis previously presented allowed the develop-
ment of an inverse analysis methodology, with the following
assumptions: (i) the experimental results under the cruciform
biaxial test, concerning the evolutions of Pvs.
D
land 
e
v
M
vs. d,
are determined in advance and (ii) the elastic properties of the
material are known. In order to illustrate the proposed inverse
analysis strategy, the ‘‘experimental’’ results are computer gener-
ated using a material, which behaviour is described by Drucker’s
yield criterion extended to anisotropy by means of a linear trans-
formation (Drucker + L) and Swift isotropic work-hardening law.
The constitutive models parameters to be identified concern
Hill’48 yield criterion and Swift work-hardening law. The Hill’48
anisotropy parameters Land Mare set equal to 1.5.
The use of computer generated results instead of experimental
ones is a simple and efficient way to test the inverse analysis meth-
odology, since the behaviour of the tested material is properly
defined, without the errors generally associated with experimental
measurements. The disadvantage of this approach is that parame-
ters identification is known to be sensitive to experimental noise,
although experimental full-field strain measurements, with noise
in the range of DIC admissible uncertainties, allow obtaining
parameters dispersion that remains in narrow and acceptable
ranges (e.g. [18]). However, the use of computer generated results
allows the suitable comparison between inverse analysis and
‘‘experimental’’ results, concerning the yield surface and the
work-hardening functions that, for real experimental cases, can
only be accessed with resource to other methodologies such as
the classical ones, using constitutive models that not adequately
describe the behaviour of the material. In fact, it is legitimate to
consider that any constitutive model cannot perfectly describe
the behaviour of a given material.
The proposed inverse analysis methodology consists on deter-
mining a primary solution, which can be enhanced using a gradi-
ent-based algorithm, i.e. the Levenberg–Marquardt method.
3.2.1. Primary solution
The initial part of the proposed inverse analysis methodology is
summarised in Table 2. This methodology is detached in six steps;
106 P.A. Prates et al. / Computational Materials Science 85 (2014) 102–120

for starting the inverse analysis (Step 1), an initial set of parame-
ters is chosen. In each of the following steps, numerical simulations
are performed and the results are compared with the experimental
ones, for verifying whether it is necessary to repeat each step of the
procedure. Table 2 shows the type of results, numerical and exper-
imental, which are compared at each step of the analysis, along the
0xand 0yaxes, as well as the variables analysed from each type of
results. The sequence shown in Table 2 does not require returning
to previous steps, i.e. the variables analysed in a given step remain
stable in the following steps, as it was concluded from a compre-
hensive study concerning the ordering of steps. The parameters
analysed and the objectives of each step are next described.
- Step 1: The initial estimate of the parameters of Hill’48 yield
criterion and Swift work-hardening law. A comprehensive
study showed that the accuracy of the final results is not influ-
enced by the first estimate, although it may influence the num-
ber of iterations. To accomplish this step, it is suggested to
determine the initial Hill’48 parameters based on the knowl-
edge of the anisotropy coefficients at angles of 0°,45°and 90°
between the tensile axis and the rolling direction of the sheet,
r
0
,r
45
and r
90
, respectively. The anisotropy coefficient is defined
as the width to thickness strains ratio in the uniaxial tensile
test. The r
0
,r
45
and r
90
values can be determined by the tradi-
tional method, using tensile tests, for example. Alternatively,
the values of r
0
and r
90
can be estimated from the results of
the material under the cruciform biaxial test, through the
following equation:
r
0
;r
90
¼ð
e
2
=
e
1
Þ=ð
e
2
=
e
1
þ1Þð5Þ
which considers that the values of r
0
and r
90
are calculated from
e
1
and
e
2
, at the 0xand 0yaxes, respectively, for a dvalue such as
the ratio
e
2
/
e
1
attains its minimum value, which naturally occurs
in the arms of the specimen. For this dvalue, the strain path is
close to uniaxial tension. The first estimate of r
45
can be
assumed, for example, as the arithmetic average between the
first estimate of the r
0
and r
90
values. For Hill’48 yield criterion,
the equations which relate Hill’48 anisotropy parameters with
the anisotropy coefficients r
0
,r
45
and r
90
are as follows:
r
0
¼H=Gr
45
¼N=ðFþGÞ0:5r
90
¼H=Fð6Þ
Another alternative for the initial values of Hill’48 parameters is
to consider their values as in isotropy, i.e. F=G=H= 0.5 and
N= 1.5.The first estimate of Swift work-hardening law parame-
ters consists on adopting values typical for the material in study.
As alternative, the first estimate of Swift work-hardening param-
eters can be obtained using a tensile test with any axis orienta-
tion relatively to the rolling direction of the sheet, or from the
cruciform biaxial test following the strain and stress values
during loading, at a point in the central region of the arm (along
the 0xaxis, for example), for which the strain path is close to
uniaxial tension.
- Step 2: The estimate of the nvalue by adjusting the slope of the
d
P
vs.
D
lresults. This slope is defined as /¼ð1=pÞP
p
i¼1
jd
P
jjd
P
j
M

i
, where |d
P
| is the absolute value of the relative dif-
ference in loading for the measuring loading point i(which cor-
responds to a certain
D
lvalue - see forward analysis) and
jd
P
j
M
¼ð1=pÞP
p
i¼1
jd
P
j
i
is the average of the absolute values of
the relative difference in loading, where pis the total number
of measuring loading points. The estimated nvalue is obtained
when the average value, /
avg
=(/(0x)+/(0y))/2, of 0x(/(0x))
and 0y(/(0y)) axes, reaches a minimum. In other words, for
the estimated nvalue, the experimental and numerical Pvs.
D
lcurves, obtained for the axis 0x(and 0y), can be superim-
posed, as much as possible, by making a vertical displacement,
proportional to the load, of one of them. Moreover, the evolu-
tion of d
P
(along both 0xand 0yaxes) during the test allows
to decide if increments or decrements of nshould be performed,
in agreement with the forward analysis. If d
P
increases with the
increase of
D
l, then decrements of nmust be performed; other-
wise, if d
P
decreases with the increase of
D
l, then increments of
nmust be performed.
- Step 3: The estimate of the (F+H)
1/2
value by reducing the dis-
agreement of the d
P
vs.
D
lresults between the 0xand 0yaxes.
Generally, at the end of the previous step, the values of the rel-
ative vertical displacements necessary for superimposing the
experimental and numerical Pvs.
D
lcurves are not equal for
the two axes, 0xand 0y. The average of the relative difference
in loading is evaluated as d
M
P
¼ð1=pÞP
p
i¼1
d
P
i
, for the 0x
(d
M
P
ð0xÞ) and the 0yaxes (d
M
P
ð0yÞ), where pis the total number
of measuring loading points. The disagreement is quantified
by the difference between the 0xand 0yresults of d
M
P
,
d
M
P
ð0x0yÞ¼ðd
M
P
ð0xÞd
M
P
ð0yÞÞ. The minimisation of
d
M
P
ð0x0yÞis achieved by acting on the (F+H)
1/2
value (for
example, by changing For H, separately). In this regard, decre-
ments of (F+H)
1/2
must be performed if d
M
P
ð0yÞ<d
M
P
ð0xÞ, other-
wise, increments of (F+H)
1/2
should be performed, in
agreement with the forward analysis. At the end of this step,
an equation (F+H)
1/2
=kcoupling the Fand Hconstitutive
parameters is established, in addition to the equation G+H=1.
- Step 4: The estimate of the F,Gand Hvalues which
minimise the average of the relative differences in von Mises
equivalent strain between the 0xand the 0yaxes,
jd

e
v
M
j
avg
¼ðjd

e
v
M
j
M
ð0xÞþjd

e
v
M
j
M
ð0yÞÞ=2. This step keeps the
(F+H)
1/2
=kvalue as adjusted in Step 3 and the condition
G+H= 1 unchanged. The average relative difference in equiva-
lent strain, defined as jd

e
v
M
j
M
¼ð1=qÞP
q
i¼1
jd

e
v
M
j
i
, is evaluated for
the 0xjd

e
v
M
j
M
ð0xÞ

and 0yjd

e
v
M
j
M
ð0yÞ

axes, where iis the
measuring point of the equivalent strain (which corresponds
to a certain dvalue – see forward analysis) and qis the total
number of points for measuring the equivalent strain. The pro-
posed approach consists in performing simulations for different
sets of F,Gand H. This is performed by changing either For H,
for example. Increments, or decrements, of F(or H) should be
performed according to the sign of the value of d

e
v
M
in the centre
of the cruciform specimen. In fact, according with the forward
analysis, if this value is positive, then decrements of F(or incre-
ments of H) should be performed; otherwise, if this value is
negative, then the opposite should be performed. Moreover, if
the d

e
v
M
value in the centre of the specimen is close to zero, sim-
ulations with both increasing and decreasing values of F(or H)
should be performed.
- Step 5: The estimate of the Nvalue which minimises jd

e
v
M
j
avg
,
while keeping unchanged the F,Gand Hvalues previously iden-
tified in Step 4. The sign of the value of d

e
v
M
in the centre of the
Table 2
Summary of the inverse analysis methodology for identifying the parameters of the
Hill’48 criterion and the Swift work-hardening law.
Step Parameters to
optimise
Results compared (Numerical –
Experimental)
Variables
analysed
1. Initial estimate of the parameters of the Swift work-hardening law and
the Hill’48 yield criterion
2. nPvs.
D
l/
avg
3. (F+H)
1/2
(=
r
0
/
r
90
)
Pvs.
D
ld
M
P
ð0x0yÞ
4. F,G,H
e
v
M
vs. djd

e
v
M
j
avg
5. N
e
v
M
vs. djd

e
v
M
j
avg
6. Y
0
Pvs.
D
ld
avg
P
P.A. Prates et al. / Computational Materials Science 85 (2014) 102–120 107

cruciform specimen allows deciding if increments, or decre-
ments, of Nshould be performed. If this value is positive, then
decrements of Nshould be performed; otherwise, if this value
is negative, then the opposite should be performed. Moreover,
if the d

e
v
M
value in the centre of the specimen is close to zero,
then simulations with both increasing and decreasing values
of Nshould be performed.
- Step 6: The estimate of Y
0
in such way that the experimental
and numerical loading curves coincide as much as possible. In
this step, the average of the relative differences in loading
between the 0xand the 0yaxes, d
avg
P
¼ðd
M
P
ð0xÞþd
M
P
ð0yÞÞ=2is
used for adjusting the Y
0
value, which is obtained when
minimising d
avg
P
. Increments of Y
0
must be performed if d
avg
P
is
negative; otherwise, decrements of Y
0
must be performed.
Briefly, the variables analysed during this exploratory inverse
analysis (see Table 2) allow converting each evolution and
distribution, such as in Fig. 2, into a unique value, making the opti-
misation procedure easier. After the initial estimate of Swift work-
hardening law and Hill’48 yield criterion parameters (Step 1), the
aim of Step 2 is to make, as much as possible, the overlapping of
the numerical and experimental Pvs.
D
l, unless a vertical displace-
ment, proportional to the load, of one of them to the other. In Step
3, the difference in relative difference in loading between the 0x
and 0yaxes is minimised. Step 4 and Step 5 concern the minimisa-
tion of the numerical and experimental 
e
v
M
vs. dresults, along both
axes. The purpose of Step 6 is to overlap the numerical and exper-
imental Pvs.
D
lcurves.
The procedure indicated in Table 2 can be extended to other con-
stitutive models provided that the analysed variables and the com-
pared results at each step remain the same. The difference is in the
parameter(s) to be optimised in each step of the procedure. That is
to say, for a given constitutive model, the parameters to be opti-
mised are the ones associated to: (Step 2) the work-hardening
(e.g. for Voce work-hardening law [34], the parameters are the
stress saturation rate and the saturation value); (Step 3) the
r
0
=
r
90
ratio; (Step 4) the normal stress components
r
xx
,
r
yy
; (Step
5) the shear stress component
s
xy
and (Step 6) the yield stress value.
3.2.2. Enhanced solution
Afterwards, the gradient-based Levenberg–Marquardt method
[35] was adopted to enhance the primary solution, provided by
the inverse analysis methodology summarised in Table 2. This
method requires an initial solution somewhat close to the final
one, otherwise, it may cause convergence problems. In this con-
text, the primary solution can be used as first solution for the min-
imisation problem, described by the following least squares cost
function:
FðAÞ¼1
qX
q
i¼1

e
exp
v
M

e
num
v
M
ðAÞ

e
exp
v
Mi
!
0x
2
þ1
qX
q
i¼1

e
exp
v
M

e
num
v
M
ðAÞ

e
exp
v
Mi
!
2
0y
þ1
pX
p
i¼1
P
exp
P
num
ðAÞ
P
exp i

2
0x
þ1
pX
p
i¼1
P
exp
P
num
ðAÞ
P
exp i

2
0y
ð7Þ
(b)(a)
(d)(c)
0x 0y
0
2
4
6
8
10
12
[mm]lΔ
[kN]P
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
[mm]d
vM
ε
-1.00
-0.50
0.00
0.50
1.00
1.50
2.00
[mm]d
21
εε
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0123456 0 9 18 27 36 45 54
0 9 18 27 36 45 54 -0.3 -0.2 -0.1 0 0.1
1
ε
2
ε
Fig. 3. Cruciform test results concerning the reference material, along the 0xand 0yaxes: (a) Pvs.
D
l; (b) 
e
v
M
vs. d; (c)
e
2
=
e
1
vs. dand (d)
e
1
vs.
e
2
.
Table 3
Initial estimate for the constitutive parameters of the Hill’48 yield criterion and the
Swift work-hardening law.
Hill’48 yield parameters Swift work-hardening parameters
FGHNY
0
(MPa) C(MPa) n
0.5000 0.5000 0.5000 1.5000 100.00 288.54 0.200
108 P.A. Prates et al. / Computational Materials Science 85 (2014) 102–120

where F(A) is the cost function; Ais the set of constitutive param-
eters to be optimised; 
e
exp
v
M
and 
e
num
v
M
are the experimental and
numerical values of von Mises equivalent strain, respectively, along
the 0xand 0yaxes, at an instant previous to maximum load; P
exp
and P
num
are the experimental and numerical load values during
the test, respectively, along both 0xand 0yaxes. In this work,
A¼½Y
0
;n;C;F;G;H;N
T
and so the value of e
0
is left free; also the
condition G+H= 1 is no longer imposed. The previous cost function
is minimised by applying Levenberg–Marquardt method that re-
quires the knowledge of Jacobian matrix, which defines the gradient
of the numerically computed measures of equivalent strain and
loads with respect to the constitutive parameters. Typically, this
matrix is updated at each iteration step k, in order to improve the
convergence. However, its calculation in each step requires high
computational cost (at least one numerical simulation of the biaxial
tensile test per constitutive parameter).
In the case in study, Jacobian matrix is computed using a for-
ward finite differentiation approach. Since the initial values for A
are obtained using the inverse analysis methodology previously
mentioned, assuming that they are close to the local minimum,
Jacobian matrix was kept unchanged during the optimisation pro-
cedure. Also, close to the local minimum, it is preferable to use a
small damping factor k, as it provides near quadratic convergent
steps towards the solution. In this work, the recommended value
of k¼kFðAÞk
2
was used [36].
3.2.3. Final remarks
In summary, the optimisation procedure uses results concern-
ing the loading curve (Pvs.
D
l) and the evolution of 
e
v
M
vs. d; this
last evolution is determined at a fixed
D
lvalue, the closer as pos-
sible to the one attained at the maximum load. Alternatively, the
evolution of 
e
v
M
vs. dcould be followed and compared at several
loading values, during the test, as performed by other authors
[19]. However, such a procedure would hinder its practical applica-
tion and proves to be unnecessary, given the accuracy obtained
when using only one
D
lvalue for measuring the strain evolution
along the 0xand 0yaxes combined with the loading curves, as
we will see later in this study. It will be also shown that the strain
measurement on points placed along the 0xand 0yaxes (which can
be performed by DIC, but also using other classical measurement
procedures) is enough for performing the accurate identification
of the constitutive parameters, without requiring the measuring
of the whole strain field of the specimen (see e.g. [19,20]), which
is a relatively complex procedure. Moreover, when comparing with
the classical methodologies, the currently developed inverse anal-
ysis strategy only requires the measuring of the load curves and
strain evolutions along both axes, avoiding the hard and fastidious
analysis of tensile, bulge and shear tests (for example), for deter-
mining the yield stresses, anisotropy coefficients and stress–strain
curve.
3.3. Inverse analysis: case study
In order to exemplify the above described inverse analysis strat-
egy, results of the cruciform biaxial test were computer generated
for a material with a plastic behaviour described by Drucker + L cri-
terion and Swift work-hardening law.
Drucker + L is a criterion for orthotropy, given by the equation
[37]:
½ð1=2Þtrðs
2
Þ
3
c½ð1=3Þtrðs
3
Þ
2
¼27ðY=3Þ
6
ð8Þ
(b)(a)
0x 0y
-50
-40
-30
-20
-10
0
[%]
P
δ
[mm]lΔ
-100
0
100
200
300
400
01234 0 10203040
vM
[%]
ε
δ
[mm]d
Fig. 4. Relative difference between the first estimate and the reference results, along the 0xand 0yaxes: (a) d
P
vs.
D
land (b) d

e
v
M
vs. d.
Table 4
Values of nused in the simulations during the second step of the inverse analysis
strategy, and the respective /values. The last line shows the results of /for the
estimated nvalue, at the end of Step 2.
n/(0x) (%) /(0y) (%) /
avg
(%)
Starting point 0.200 2.79 2.70 2.75
Trials 0.220 2.01 1.99 2.00
0.240 1.23 1.28 1.26
0.260 0.46 0.56 0.51
0.280 0.68 0.53 0.61
0.300 1.59 1.29 1.44
Final estimate 0.268 0.48 0.36 0.42
y = 987.5x2- 530x + 71.57
R² = 0.9992
0.00
0.50
1.00
1.50
2.00
2.50
3.00
0.20 0.22 0.24 0.26 0.28 0.30
n
avg
[%]
φ
Fig. 5. Polynomial fitting of the results in Table 4, concerning /
avg
as a function of n
(open symbols); the solid symbol represents the estimated value.
P.A. Prates et al. / Computational Materials Science 85 (2014) 102–120 109

where tr(s) is the trace of the stress tensor s, resulting from the lin-
ear transformation of the effective Cauchy stress tensor,
r
, and cis a
weight parameter, ranging between 27/8 and 9/4, to ensure the
convexity of the yield surface. When cequals zero, this criterion
coincides with Hill’48 yield criterion. The sstress tensor is given by:
s¼L:
r
ð9Þ
where Lis the linear transformation operator, written as follows:
L¼
ðC
2
þC
3
Þ=3C
3
=3C
2
=3000
C
3
=3ðC
3
þC
1
Þ=3C
1
=3000
C
2
=3C
1
=3ðC
1
þC
2
Þ=30 0 0
000C
4
00
0000C
5
0
00000C
6
2
6
6
6
6
6
6
6
6
4
3
7
7
7
7
7
7
7
7
5ð10Þ
where C
i
, with i=1,..., 6, are the anisotropy parameters. This yield
criterion includes one more parameter, the parameter c, than Hill’48
yield criterion, thus being more flexible. So, when the parameter cis
not zero, Hill’48 criterion cannot fully describe the behaviour of a
material that follows Drucker + L criterion.
The purpose of this case study is to find the set of parameters
for Hill’48 yield criterion and Swift work-hardening law, which
best describe the reference results of the cruciform test, considered
as experimental. In order to obtain these results, the anisotropy
parameters considered for Drucker + L criterion are C
1
= 0.6681,
C
2
= 0.8158, C
3
= 1.2394, C
4
=C
5
= 1.0000, C
6
= 0.9440, c= 1.4265
and Swift law parameters are Y
0
= 118.63 MPa, C= 502.61 MPa
and n= 0.268. The elastic properties are: Young’s modulus,
E= 210 GPa and Poisson ratio,
m
= 0.3. Fig. 3 shows the numerically
generated results of the cruciform test, along both axes. Fig. 3(a)
shows the Pvs.
D
levolution; Fig. 3(b) shows the 
e
v
M
vs. ddistribu-
tion; Fig. 3(c) present the
e
2
/
e
1
vs. ddistribution and Fig. 3(d) the
e
1
vs.
e
2
distribution. The results in Fig. 3(b–d) were obtained after
displacements of the specimen boundaries
D
lequal to 6 mm, pre-
ceding the maximum load along the 0xand 0yaxes. In order to
compare material results with the ones obtained at the different
steps of the inverse analysis, the number of points used to evaluate
the loading evolution and the equivalent strain distribution, pand
q, respectively, are equal to 100 for each axis.
Table 3 shows the parameters of Hill’48 criterion (as for isot-
ropy), as well as Swift work-hardening parameters, which were
estimated based on values typical for the material in study (which
is somewhat identical to a mild-steel), used as initial estimate of
(b)
(a)
0x 0y
-50
-40
-30
-20
-10
0
[%]
P
δ
[mm]lΔ
-100
0
100
200
300
400
0123456 010203040
vM
[%]
ε
δ
[mm]d
Fig. 6. Relative difference results concerning the second estimate: (a) d
P
vs.
D
land (b) d

e
v
M
vs. d.
Table 5
Values of ðFþHÞ1=2used in the simulations during the third step of the inverse analysis strategy, and the respective dM
Pvalues. The last line shows the estimated ðFþHÞ1=2value
for each approach (For Happroaches, in square brackets) and the corresponding dM
Presults.
FH
(F+H)
1/2
d
M
P
ð0xÞ(%) d
M
P
ð0yÞ(%) d
M
P
ð0x0yÞ(%) /
avg
(%) d
M
P
ð0xÞ(%) d
M
P
ð0yÞ(%) d
M
P
ð0x0yÞ(%) /
avg
(%)
Starting point 1.0000 22.75 29.26 6.51 0.42 22.75 29.26 6.51 0.42
Trials 0.9747 22.55 26.72 4.17 0.41 23.76 27.89 4.13 0.49
0.9487 22.33 23.93 1.60 0.39 25.60 27.20 1.60 1.40
0.9220 22.11 20.87 1.24 0.37 26.74 25.74 1.00 1.50
0.8944 21.88 17.47 4.41 0.35 28.05 24.37 3.68 0.50
Final estimate 0.9358 [Fapproach] 22.23 22.48 0.25 0.38
0.9329 [Happroach] 25.52 25.60 0.08 0.56
y = -103.34x + 96.639
R² = 0.9981
y = -96.685x + 90.141
R²= 1
-8
-6
-4
-2
0
2
4
6
0.85 0.90 0.95 1.00 1.05 1.10
F
H
12
()FH+
M
(0x 0y)[%]
P
δ
−
Fig. 7. Linear fitting of the results in Table 5, concerning d
M
P
ð0x0yÞas a function of
ðFþHÞ
1=2
, regarding the For Happroaches (open symbols); the solid symbols
represent the estimated values for each case.
110 P.A. Prates et al. / Computational Materials Science 85 (2014) 102–120

the inverse analysis procedure (Step 1). In Table 3, the parameter
e
0
, of Swift law, is equal to 0.005, which is kept unchanged during
the inverse analysis, until determination of the primary solution.
According to Table 2, steps 2, 3 and 6 are focused on the analysis
on the d
P
vs.
D
levolution while steps 4 and 5 use the d

e
v
M
vs. d
distribution. Nevertheless, to make it easy to understand the pro-
gression of the optimisation procedure, both types of results are
shown in all steps.
The second step of the inverse analysis strategy (see Table 2)
started by performing a numerical simulation, considering the ini-
tial estimate of the constitutive parameters presented in Table 3,
and comparing the results with the experimental ones. Fig. 4
shows the results of the relative difference of the loading evolu-
tions (Fig. 4(a)) and von Mises equivalent strain distributions
(Fig. 4(b)), along both 0xand 0yaxes. Results in Fig. 4(a) are dis-
played up to a tool displacement
D
l= 4 mm, since the maximum
load value obtained for the initial estimate occurs for a displace-
ment value much less than 6 mm, observed for the experimental
results (see Fig. 3(a)); consequently, Fig. 4(b) was plotted for
D
l= 4 mm.
Fig. 4(a) shows a gradual increase on the negative side of the
relative difference throughout the test, corresponding to a value
of 2.74 for /ð0xÞand 2.66 for /ð0yÞslopes. The forward analysis
revealed that an increase of nis related with a gradual increase
on the positive side of the evolution of the relative load difference,
during the test. Therefore, in Step 2, five trial simulations were per-
formed in parallel, considering increasing values of n, while keep-
ing the remaining parameters unchanged. Table 4 shows the
results of /ð0xÞ,/ð0yÞand /
avg
, for these simulations (i.e. with
increasing work-hardening exponent values). A 2nd order polyno-
mial fit of the results of /
avg
as a function of nwas performed, as
shown in Fig. 5; a work-hardening exponent of n= 0.268 was
estimated from the minimum value of this fit. A new numerical
simulation was performed using this nvalue, in order to determine
the respective values of /ð0xÞand /ð0yÞ(see Table 4 and Fig. 5).
The results obtained were compared with the experimental ones,
as shown in Fig. 6(a) for the relative difference results in loading
and in Fig. 6(b) for von Mises equivalent strain, along both 0x
and 0yaxes. Results in Fig. 6(a) are displayed up to a load value
prior to the maximum, i.e. for a tool displacement of 6 mm (the
maximum load appears after this displacement value for both
the numerical and experimental results); consequently, Fig. 6(b)
was obtained for
D
l= 6 mm. In general, Fig. 6(a) results display a
steady value for the relative difference in loading, during the test,
as a result of the improved estimation of n. Therefore, the obtained
/
avg
value equal to 0.42 was considered small enough to proceed to
the next step.
The third step of the inverse analysis strategy (see Table 2)
consists in performing trial numerical simulations with different
values of (F+H)
1/2
, by either changing For H, in order to find the
(F+H)
1/2
value that minimises d
M
P
ð0x0yÞ, i.e. minimises the dis-
agreement between the average relative difference in loading
results for both axes. Forward analysis showed that when
ðFþHÞ
1=2
increases, the relative difference of the loading results
along the 0yaxis becomes located below to the results obtained
for the 0xaxis, i.e. d
M
P
ð0yÞ<d
M
P
ð0xÞ(see Fig. 2(b)). Therefore, trial
simulations with decreasing values of ðFþHÞ
1=2
were performed
(b)
(a)
0x [F] 0y [F] 0x [H] 0y [H]
-40
-30
-20
-10
0
10
[%]
P
δ
[mm]lΔ
-100
100
300
500
700
900
0123456 010203040
[mm]d
vM
[%]
ε
δ
Fig. 8. Relative difference results concerning the third estimate, along both 0xand 0yaxes, for each approach (Fand Happroaches, in square brackets): (a) d
P
vs.
D
land (b)
d

e
v
M
vs. d.
Table 6
Estimate of Ffrom the fourth step of the inverse analysis strategy (the value of
ðFþHÞ1=2is kept unchanged and the condition G+H= 1 is preserved).
Fjd

e
v
M
j
M
ð0xÞ
(%)
jd

e
v
M
j
M
ð0yÞ
(%)
jd

e
v
M
j
avg
(%)
d
M
P
ð0x0yÞ
(%)
/
avg
(%)
Starting point 0.3757 116.48 106.36 111.42 0.25 0.38
Trials 0.3000 55.54 50.31 52.93 0.14 0.23
0.2500 24.65 21.95 23.30 0.05 0.17
0.2000 4.80 4.65 4.73 0.07 0.23
0.1500 19.52 18.51 19.02 0.20 0.25
0.1000 33.78 31.82 32.80 0.36 0.38
Final estimate 0.1967 4.44 4.44 4.44 0.07 0.24
y = 6573x2-
2586.3x + 259.07
0
20
40
60
80
100
120
0.00 0.10 0.20 0.30 0.40
F
vM
avg [%]
ε
δ
Fig. 9. Polynomial fitting of the results jd

e
v
M
j
avg
as a function of F(open symbols).
The solid symbol represents the estimated Fvalue and the obtained jd

e
v
M
j
avg
value.
P.A. Prates et al. / Computational Materials Science 85 (2014) 102–120 111

in parallel, during this step. Table 5 shows the results of d
M
P
ð0xÞ,
d
M
P
ð0yÞand d
M
P
ð0x0yÞobtained from two possible approaches,
i.e. by either decreasing of For H.Fig. 7 shows the estimated
ðFþHÞ
1=2
value based on these two approaches by using a linear
fitting. Both approaches lead to quite similar estimations of the
ðFþHÞ
1=2
value (0.9358, when acting on F, or 0.9329, when acting
on H).
Two numerical simulations were performed using both final
estimations of the ðFþHÞ
1=2
value, in order to determine the
respective values of d
M
P
ð0x0yÞ(see Table 5 and Fig. 7). The results
obtained for d
P
vs.
D
land d

e
v
M
vs. dare shown in Fig. 8(a and b),
respectively, along both 0xand 0yaxes. Results in Fig. 8(a) are
displayed up to a load value prior to the maximum, i.e. for a tool
displacement of 6 mm; Fig. 8(b) was obtained for
D
l= 6 mm. In
general, for each final estimation, Fig. 8(a) displays an overlapping
of d
P
vs.
D
lfor the 0xand 0yaxes, as a result of the improved esti-
mation of ðFþHÞ
1=2
, for both approaches. The values of d
M
P
ð0x0yÞ
obtained from the linear fittings of Fig. 7 are 0.25 and 0.08, for the
approaches with Fand H, respectively. During this step, the vari-
ables studied in the previous step (/
avg
) almost not change its
value (/
avg
= 0.38 (for the Fapproach) and /
avg
= 0.56 (for the H
approach), in contrast with /
avg
= 0.42 for the starting point in
Table 5).
In the fourth step of the inverse analysis strategy (see Table 2)
simulations with different F,Gand Hsets are carried out, while
keeping constant the (F+H)
1/2
value found in Step 3 and the
condition G+H= 1 unchanged. This step uses as starting
point the final estimate of the parameters corresponding to
(F+H)
1/2
= 0.9358 (Fapproach in Table 5). The d

e
v
M
value in the
centre of the specimen is positive (see Fig. 8(b)), therefore, five trial
simulations with decreasing values of Fwere performed in parallel
during this step. Table 6 shows the results of jd

e
v
M
j
M
obtained for
the different Fvalues, emphasising the value F= 0.1967, retained
from a 2nd order polynomial fit on jd

e
v
M
j
avg
vs. F, as shown in
Fig. 9. The value of jd

e
v
M
j
avg
obtained from the polynomial fitting
of Fig. 9 is equal to 4.44%. Also, the variables studied in the previ-
ous steps (d
M
P
ð0x0yÞ;/
avg
) are not significantly affected with the
changing of the Fparameter. The numerical simulation results ob-
tained with the improved value of Fare compared with the exper-
imental ones in Fig. 10, which show the relative differences in
loading (Fig. 10(a)) and in von Mises equivalent strain
(Fig. 10(b)), for both axes. Fig. 10(b) display a global shifting of
the d

e
v
M
values towards 0%, when compared with the third estimate
(see Fig. 8(b)), resulting from the improved estimation of the F
parameter. Fig. 10(a) shows a slight positive vertical displacement
of the d
P
values, when compared with the third estimate corre-
sponding to the approach of decreasing F(see Fig. 8(a)).
The fifth step of the inverse analysis strategy (see Table 2) con-
sists in performing trial numerical simulations with different Nval-
ues, while keeping the other parameters unchanged, in order to
find the value that minimises the overall relative difference in
von Mises equivalent strain distribution, jd

e
v
M
j
avg
. Starting from
the estimate in Step 4, the d

e
v
M
value in the centre of the specimen
is positive, although not far from zero (5% – see Fig. 10(b)); there-
fore, four trial simulations, two of which with increasing and the
other two with decreasing values of Nare considered in this step.
From Table 7 it is shown that the jd

e
v
M
j
M
values obtained for the dif-
ferent Nvalues are at the vicinity of the minimum, namely the
(b)
(a)
0x 0y
-40
-30
-20
-10
0
10
[mm]lΔ
[%]
P
δ
-20
-15
-10
-5
0
5
10
0123456 010203040
[mm]d
vM
[%]
ε
δ
Fig. 10. Relative difference results concerning the fourth estimate: (a) d
P
vs.
D
land (b) d

e
v
M
vs. d.
Table 7
Estimate of Nfrom the fifth step of the inverse analysis strategy.
Njd

e
v
M
j
M
ð0xÞ
(%)
jd

e
v
M
j
M
ð0yÞ
(%)
jd

e
v
M
j
avg
(%)
d
M
P
ð0x0yÞ
(%)
/
avg
(%)
Starting point 1.5000 4.44 4.44 4.44 0.07 0.24
Trials 1.7000 5.93 5.93 5.93 0.02 0.24
1.6000 5.00 5.10 5.05 0.05 0.24
1.4000 4.55 4.51 4.53 0.09 0.24
1.3000 6.26 6.43 6.35 0.12 0.25
Final estimate 1.4620 4.42 4.28 4.35 0.08 0.24
y = -121.25x3+ 589.13x2- 945.13x
+ 505.76
R² = 0.9982
0
1
2
3
4
5
6
7
1.00 1.20 1.40 1.60 1.80 2.00
N
vM
avg [%]
ε
δ
Fig. 11. Polynomial fitting of the results of jd

e
v
M
j
avg
as a function of N(open
symbols). The solid symbol represents the estimated Nvalue and the obtained
jd

e
v
M
j
avg
value.
112 P.A. Prates et al. / Computational Materials Science 85 (2014) 102–120

starting point value (N= 1.5000). The value of N= 1.4620 was
retained from the polynomial fitting on the results of jd

e
v
M
j
avg
vs.
N, as shown in Fig. 11. The value of jd

e
v
M
j
avg
obtained from the poly-
nomial fitting of Fig. 11 is equal to 4.35%. Also, the variables stud-
ied in the previous steps (d
M
P
ð0x0yÞand /
avg
) are not significantly
affected with the changing of Nvalue. The numerical simulation
results obtained with the improved value of Nare compared with
the experimental ones in Fig. 12, which show the relative differ-
ences in loading (Fig. 12(a)) and von Mises equivalent strain
(Fig. 12(b)), for both axes. Fig. 12(b) display a slight improvement
of the jd

e
v
M
j
avg
values when compared with the fourth estimate (see
Fig. 10(b)), due to an improvement of the Nparameter estimation.
Fig. 12(a) shows no changes of the relative difference in loading,
when compared with the fourth estimate (see Fig. 10(a)).
The sixth step of the inverse analysis strategy (see Table 2) con-
sists in performing numerical simulations with different values of
Y
0
, in order to minimise the overall relative difference in load, d
avg
P
.
In the forward analysis it was shown that an increase of Y
0
is
related with an increase of d
avg
P
. Therefore, in this step, and taking
the estimate results of the fifth step as reference, four trial simula-
tions were performed with increasing values of Y
0
, while keeping
the remaining parameters unchanged. Table 8 shows the results ob-
tained for d
M
P
as a function of Y
0
, for all these simulations, including
the estimate from step five. Table 8 also shows the variables ana-
lysed in the previous steps, in order to show that the respective dif-
ferences are not significantly affected when changing the Y
0
value.
The yield stress equal to 122.23 MPa was estimated from a linear fit-
ting of the d
avg
P
vs. Y
0
results, as shown in Fig. 13. The value of d
avg
P
ob-
tained from the linear fitting of Fig. 13 is equal to 0.01%. Also, the
variables studied in the previous steps (jd
e
v
M
j
avg
;d
P
ð0x0yÞ;/
avg
)
indicate enough accuracy, at the end of this step. The results ob-
tained with the improved estimation of Y
0
were compared with
the experimental ones and are shown in Fig. 14: (a) the relative dif-
ference in loading and (b) von Mises equivalent strain, along both 0x
and 0yaxes. Fig. 14(a) displays values for the relative difference in
loading close to 0%, i.e. the numerical and material Pvs.
D
lcurves be-
come similar.
The inverse analysis methodology previously presented allowed
the determination of a primary solution for the constitutive param-
eters (with a total of 28 simulations). This primary solution was used
as first solution for the error minimisation problem defined in Eq.
(7), using Levenberg–Marquardt optimisation method. This method
has converged to a solution with a cost function value 20% lower
than the one obtained at the end of Step 6, after three iterations
(with a total of 10 simulations). Otherwise, using Levenberg–Mar-
quardt algorithm for the parameter identification, starting from
the initial estimate in Table 3 (Full-LM identification), a similar va-
lue of the cost function was reached after 5 iterations, but in this case
it was necessary to update Jacobian matrix in each iteration (with a
total of 41 simulations). Fig. 15 compares the relative difference in
loading (Fig. 15(a)) and von Mises equivalent strain (Fig. 15(b)),
along both 0xand 0yaxes, at the end of Step 6 (PS), after applying
Levenberg–Marquardt optimisation strategy (LM) and for full
Levenberg–Marquardt optimisation (Full-LM). Table 9 allows com-
paring the identified constitutive parameters for all cases in
Fig. 15, showing the corresponding cost function values and the val-
ues for jd

e
v
M
j
avg
;d
avg
P
;d
M
P
ð0x0yÞand /
avg
. In addition, the cost func-
tion was also evaluated for the results from Step 1 of the inverse
analysis (Initial estimate). The major reduction of F(A) and of both
d
M
P
and jd

e
v
M
j
M
occur during the inverse analysis strategy.
(b)
(a)
0x 0y
-40
-30
-20
-10
0
10
[%]
P
δ
[mm]lΔ
-20
-15
-10
-5
0
5
10
0123456 0 10203040
[mm]d
vM
[%]
ε
δ
Fig. 12. Relative difference results concerning the fifth estimate: (a) d
P
vs.
D
land (b) d

e
v
M
vs. d.
Table 8
Estimate of Y0from the sixth step of the inverse analysis strategy.
Y
0
(MPa)
d
M
P
ð0xÞ
(%)
d
M
P
ð0yÞ
(%)
d
avg
P
(%)
jd

e
v
M
j
avg
d
M
P
ð0x0yÞ
(%)
/
avg
(%)
Starting point 100.00 18.12 18.04 18.08 4.35 0.08 0.24
Trials 110.00 9.98 9.90 9.94 4.39 0.08 0.23
120.00 1.85 1.77 1.81 4.44 0.08 0.21
130.00 6.28 6.36 6.32 4.49 0.08 0.18
140.00 14.40 14.47 14.44 4.55 0.07 0.16
Final estimate 122.23 0.03 0.05 0.01 4.45 0.08 0.13
y = 0.8126x - 99.342
R² = 1
-20
-15
-10
-5
0
5
10
15
20
100 110 120 130 140
0
[MPa]Y
avg
[%]
P
δ
Fig. 13. Linear fitting of the results of d
avg
P
vs. Y
0
(open symbols). The solid symbol
represents the estimated Y
0
value and the obtained d
avg
P
value.
P.A. Prates et al. / Computational Materials Science 85 (2014) 102–120 113

According to the cost function used for this optimisation meth-
od (see Eq. (7)), the results for ‘‘LM’’ and ‘‘Full-LM’’ cases are
slightly better than the results for ‘‘PS’’. However, in the perspec-
tive of the exploratory inverse analysis methodology, the
jd

e
v
M
j
avg
;d
avg
P
and /
avg
variables (evaluated during Step 2) are lower
for ‘‘PS’’ than for ‘‘LM’’, and only the value of the variable jd

e
v
M
j
avg
is improved for the ‘‘Full-LM’’ solution, having increased the values
of the variables d
avg
P
,d
M
P
ð0x0yÞ, and /
avg
. That is, for the case of
the ‘‘Full-LM’’ solution, the gain in variable jd

e
v
M
j
avg
is compensated
by a loss in the remaining variables. In conclusion, the quality of
the results obtained by the different optimisation strategies (using
variables such as jd

e
v
M
j
avg
,d
avg
P
,d
M
P
ð0x0yÞ, and /
avg
, for the ‘‘PS’’, or
cost function, for the gradient-based optimisation strategy) cannot
be directly compared, as the quantities minimised in the optimisa-
tion strategies are distinct.
4. Classical identification strategy
In this section, the results obtained by the inverse analysis
strategy previously presented are compared with the results
obtained by a classical methodology. For this case, Hill’48 anisot-
ropy parameters are identified from computer generated results
of the following linear strain path tests: tensile and shear tests,
with an angle
a
at 0°,15°,30°,45°,60°,75°and 90°with the rolling
direction, for which were determined the yield stress and anisot-
(b)
(a)
0x 0y
-40
-30
-20
-10
0
10
[%]
P
δ
[mm]lΔ
-20
-15
-10
-5
0
5
10
0123456 0 10203040
[mm]d
vM
[%]
ε
δ
Fig. 14. Relative difference results concerning the sixth estimate: (a) d
P
vs.
D
land (b) d

e
v
M
vs. d.
(a) (b)
PS_0x LM_0x Full-LM_0x
PS_0y LM_0y Full-LM_0y
-5
0
5
10
15
20
25
[mm]lΔ
P
δ
-20
-15
-10
-5
0
5
10
0246 010203040
[mm]d
vM
[%]
ε
δ
[%]
Fig. 15. Relative difference results at the end of Step 6 (PS), after applying the Levenberg–Marquardt strategy (LM) to the primary solution and for the complete Levenberg–
Marquardt optimisation (Full-LM), along both 0xand 0yaxes: (a) d
P
vs.
D
land (b) d

e
v
M
vs. d.
Table 9
Cost function values, jd
evMjavg,davg
P,dM
Pð0x0yÞand /
avg
values and constitutive parameters for the inverse analysis at Step 1 (Initial Estimate), at the end of the Step 6 (PS), at the
end of the Levenberg–Marquardt optimisation strategy (LM) and for the full Levenberg–Marquardt optimisation (Full-LM).
F(A)jd

e
v
M
j
avg
(%) d
avg
P
(%) d
M
P
ð0x0yÞ(%) /
avg
(%) Hill’48 Anisotropy Parameters Swift Law Parameters
FGHNY
0
[MPa] C[MPa] n
Initial estimate 5.3022 116.11 36.84 5.45 2.75 0.5000 0.5000 0.5000 1.5000 100.00 288.54 0.200
PS 0.0090 4.45 0.01 0.08 0.13 0.1967 0.3210 0.6790 1.4620 122.23 505.65 0.268
LM 0.0075 4.71 0.71 0.02 0.56 0.1901 0.3213 0.6787 1.3811 134.60 527.12 0.296
Full-LM 0.0067 3.24 1.99 0.24 1.54 0.2201 0.3431 0.6569 1.0014 156.82 547.61 0.344
114 P.A. Prates et al. / Computational Materials Science 85 (2014) 102–120

ropy coefficient evolutions in the sheet plane,
r
a
and r
a
, and the
yield stress in shear,
s
a
; circular bulge and compression tests, for
determining the biaxial yield stress,
r
b
, and the biaxial anisotropy
coefficient, r
b
, respectively. These variables were determined fol-
lowing the same assumptions used in experimental analyses: the
values of the yield stresses
r
a
,
s
a
, and
r
b
were obtained from the
corresponding stress–strain curves at a plastic strain value of
0.1% [38]; the r
a
anisotropy coefficients were determined using
strain values from 0.5% up to the onset of necking [38]. As usually,
the biaxial stress–strain curve was determined from the bulge test
results assuming that the material is isotropic and with resource to
the membrane theory [39]. The r
b
value, defined by r
b
¼
e
2
=
e
1
, was
determined using strain values between 0.5% and 50%.
The parameters of Hill’48 criterion and Swift work-hardening
law were also directly identified from the equations describing
the behaviour of the material, in order to check the influence of
the uncertainties of the experimental analysis on the classical
identification results. Concomitantly, this allows to check the
influence of assumptions generally taken in the analysis of the
experimental data (for example: the yield stress defined at 0.1%
of plastic strain; the isotropy condition imposed for determining
the equivalent stress–strain curve) on the results of the parameter
identification. Fig. 16 shows the comparison between the
measured variables according to the experimental and the analyt-
ical procedures, namely the yield stress in tension and shear and
the anisotropy coefficient as a function of the angle,
a
, between
the tensile and shear directions and the rolling direction
(Fig. 16(a), (b) and (c), respectively). The classical experimental
procedure used in this section leads to different results when com-
pared with the analytical ones, concerning the yield stresses values
(Fig. 16(a and b)). However, the distribution of the anisotropy coef-
ficient, obtained from the experimental procedure (Fig. 16(c)),
shows similar results to the analytical solution. The biaxial yield
stress values obtained from the experimental procedure and ana-
lytically are 131.75 MPa and 165.85 MPa, respectively. The biaxial
anisotropy coefficient obtained from the experimental procedure is
0.483 and the analytical value is 0.476.
Subsequently, the classical identification of the parameters is
performed from the results obtained from both procedures, like
experimental and analytical. For both cases, the difference be-
tween the results and the optimisation approach is used to define
the cost function with the following formulation [40]:
FðAÞ¼Xw
i
ð
r
a
=
r
exp
a
1Þ
2
þXw
i
ðr
a
=r
exp
a
1Þ
2
þXw
i
ð
s
a
=
s
exp
a
1Þ
2
þXw
i
ð
r
b
=
r
exp
b
1Þ
2
þXw
i
ðr
b
=r
exp
b
1Þ
2
ð11Þ
(b)(a)
(c)
Experimental_Procedure Analytical_Procedure
100
110
120
130
140
150
160
[]
α
°
[MPa]
α
σ
62
64
66
68
70
72
74
[]
α
°
[MPa]
α
τ
0
1
2
3
4
5
6
7
0 153045607590 0 153045
0 153045607590
r
α
[]
α
°
Fig. 16. Comparison between the evolutions of the measured variables with
a
, according to the experimental and the analytical procedures: (a) initial yield stress in tension
and (b) in shear, (c) anisotropy coefficient.
Table 10
Constitutive parameters identified from inverse analysis, i.e. primary solution (PS)
and after Levenberg–Marquardt optimisation (LM), from the full Levenberg–Marqu-
ardt optimisation (Full-LM) and using the classical methodology with both proce-
dures, i.e. like experimental and analytical.
Hill’48 anisotropy parameters Swift law parameters
FGHNY
0
(MPa) C(MPa) n
PS 0.1967 0.3210 0.6790 1.4620 122.23 505.65 0.268
LM 0.1901 0.3213 0.6787 1.3811 134.60 527.12 0.296
Full-LM 0.2201 0.3431 0.6569 1.0014 156.82 547.61 0.344
CM_E 0.1819 0.3877 1.0329 1.6725 135.70 553.89 0.268
CM_A 0.1597 0.3397 0.9258 1.4935 128.38 555.80 0.268
P.A. Prates et al. / Computational Materials Science 85 (2014) 102–120 115

Arepresents the set of anisotropy parameters for the yield criterion,
r
exp
a
,r
exp
a
,
s
exp
a
,
r
exp
b
and r
exp
b
are the experimental values,
r
a
,r
a
,
s
a
,
r
b
and r
b
are the corresponding values predicted by the constitutive
equations and w
i
are weighting factors (in this study, w
i
is equal
to 1 for all the performed tests). This objective function was mini-
mised using the downhill simplex method [40]. Swift law parame-
ters were identified assuming that the equivalent work-hardening
curve is the average of the seven stress–strain curves in tension,
plotted in agreement with von Mises definition of stress and strain
[38].
(b)
(a)
(d)(c)
Mat PS LM Full-LM CM_E CM_A
100
150
200
250
300
350
p
0
ε
=
p
0.1
ε
=
[MPa]
α
σ
[]
α
°
-1.0
-0.8
-0.6
-0.4
-0.2
0.0
21
εε
[]
α
°
-500
-250
0
250
500
p
0.1
ε
=
p
0
ε
=
yy
[MPa]
σ
xx
[MPa]
σ
0
100
200
300
400
0153045607590 0 153045607590
-500 -250 0 250 500 0 0.05 0.1 0.15 0.2
100
120
140
160
180
00.01
[MPa]
Y
p
ε
Fig. 17. Comparison between the material (Mat) and identified results: (a) yield stress in tension as a function of a, for 
e
p
= 0 (lower curves) and 
e
p
= 0.1 (upper curves); (b)
tensile strain path, distributions as a function of
a
; (c) the yield surfaces, for 
e
p
= 0 (inner curves) and 
e
p
= 0.1 (outer curves) and (d) equivalent stress–strain curves. The
identified results concern the primary solution (PS), after applying the Levenberg–Marquardt optimisation to the primary solution (LM), the complete Levenberg–Marquardt
optimisation (Full-LM) and for the classical identification methodologies, with both procedures, like experimental (CM_E) and analytical (CM_A).
εε
21
εε
-1.5
-1.0
-0.5
0.0
0.5
1.0
0 330 60
(b)
[md
90
mm]
1
]
120
α
α
α
150
α
=
0
α
=
4
α
=
9
0
º
4
5º
9
0º
(a)
Fig. 18. (a) Projection of the deformed shape (1/4), on the 0xy plane; (b)
e
2
=
e
1
vs. dfor the material case, along the 0x(
a
=0°), 0x=0y(
a
=45°) and 0y(
a
=90°) axes (
e
2
and
e
1
are the minor and major strains in the sheet plane).
116 P.A. Prates et al. / Computational Materials Science 85 (2014) 102–120

The sets of parameters identified by inverse analysis with and
without using Levenberg–Marquardt optimisation and by both
procedures using the classical identification strategy are presented
in Table 10.Fig. 17 shows the yield stress (initial, i.e. for 
e
p
¼0, and
after a certain amount of equivalent plastic strain, 
e
p
¼0:1) and the
strain path in tension, defined by the ratio
e
2
=
e
1
, with
a
(Fig. 17(a
and b), respectively), the yield surfaces, for 
e
p
¼0 and for 
e
p
¼0:1,
in the
r
xx

r
yy
plane (Fig. 17(c)), and the equivalent stress–strain
curves, with a detail of the beginning of plastic deformation
(Fig. 17(d)). These results are shown for the material (Mat), for
the identifications obtained by inverse analysis strategy, i.e. the
primary solution (PS) and after Levenberg–Marquardt optimisation
(a)
(c)(b)
Mat PS LM Full-LM CM_E CM_A
25
30
35
40
45
50
55
60
[]
α
°
[mm]s
0
40
80
120
160
200
[mm]lΔ
[kN]P
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0 153045607590
0204060 0 20 40 60 80 100 120
[mm]d
vM
ε
(e)(d)
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
[mm]d
vM
ε
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0 50 100 150 200 0 20 40 60 80 100 120
[mm]d
vM
ε
Fig. 19. Cross-shaped cup deep-drawing results: (a) flange draw-in (s) as a function of
a
, (b) Pvs.
D
l, (c), (d) and (e) 
e
v
M
vs. d, along the 0x,0x=0yand0yaxes, respectively, for
the material case (Mat), primary solution of the inverse analysis (PS), inverse analysis after the Levenberg–Marquardt optimisation (LM), complete Levenberg–Marquardt
optimisation (Full-LM) and classical identification methodology, like experimental and analytical (CM_E and CM_A, respectively).
P.A. Prates et al. / Computational Materials Science 85 (2014) 102–120 117

strategy (LM), for full Levenberg–Marquardt optimisation (Full-
LM) and for the identifications based on the classical methodology,
as experimentally (CM_E) and analytically (CM_A).
When comparing the results from the identified sets of param-
eters to the material ones, the following is observed: (i) for 
e
p
¼0,
the yield stress distribution with
a
(Fig. 17(a)) is best described by
both classical identifications and for the primary solution of the in-
verse analysis, and is overestimated for both cases identifications,
‘‘LM’’ and ‘‘Full-LM’’; moreover, for 
e
p
¼0:1, the experimental
results are best described by ‘‘PS’’, ‘‘LM’’, ‘‘Full-LM’’ and ‘‘CM_A’’
cases; (ii) the distribution of the tensile strain path with
a
(Fig. 17(b)) is better described by both classical identifications than
by ‘‘PS’’, ‘‘LM’’ and ‘‘Full-LM’’; (iii) the yield surfaces (see Fig. 17(c))
obtained from both classical identifications (with experimental
and analytical analysis) and the primary solution of the inverse
analysis identification seem to better describe the material initial
yield surface than both identifications using Levenberg–Marquardt
algorithm; for 
e
p
¼0:1, the yield surfaces obtained from both in-
verse analysis identifications (primary solution, ‘‘PS’’, and after
Levenberg–Marquardt optimisation, ‘‘LM’’) well fit the material
yield surface (these results are in agreement with the results of
the yield stress distribution in the sheet plane (Fig. 17(a))), and
(iv) the equivalent stress–strain curve obtained from the inverse
analysis identification for the primary solution has an overall good
description of the material curve; in contrast, the cases ‘‘LM’’ and
‘‘Full-LM’’ overestimate the material curve only until 1% and 2.5%
of deformation, respectively, and both classical identification strat-
egies overestimate the entire material curve. In conclusion, cases
‘‘PS’’, ‘‘LM’’ and ‘‘Full-LM’’ allows closely describing the work-hard-
ening behaviour (Fig. 17(a), (c) and (d)), at least for strain values
larger than 1% (LM) and 2.5% (Full-LM), whereas the classical strat-
egies mainly describe the distributions of the tensile strain path, in
the sheet plane (Fig. 17(b)).
5. Deep-drawing of a cross-shaped cup
The deep drawing of a cross-shaped cup (see Fig. 18(a)) is used
to support the proposed inverse analysis strategy, since this pro-
cess exhibit high plastic strain values and deformation heterogene-
ity, with a wide range of strain paths, as shown in Fig. 18(b). In this
section, the numerical simulations performed use the parameters:
(i) of the material, (ii) obtained by the proposed inverse analysis
strategy, with and without the final optimisation, (iii) obtained
by a full Levenberg–Marquardt optimisation and (iv) obtained by
a classical identification strategy with both sets of like experimen-
tal and analytical data. The deep drawing process was simulated
taking into account the tool geometry, which includes a closed
die and a total punch displacement of 60 mm. Due to geometric
and material symmetries, only one quarter of the cup geometry
was considered. The blank, with initial dimensions
125 125 1mm
3
, was discretised with tri-linear 8-node hexa-
hedral solid elements with an average in-plane size of 2.0 mm
and two layers through-thickness. The blank holder applies a total
force of 290 kN and the frictional contact between the sheet and
the tools is modelled by Coulomb law, with a friction coefficient
equal to 0.03 [41].
Fig. 18(a) displays the deformed shape of the simulated part
after the deep-drawing test for the material (Mat) on the initial
plane of the sheet, 0xy;Fig. 18(b) shows the complex strain path
distribution along the 0x(
a
=0°), 0x=0y(
a
=45°) and 0y
(
a
=90°) axes for the material case.
Fig. 19(a) allows comparing the results of the draw-in (s) of the
flange as a function of
a
(as shown in Fig. 18(a)) for the material
with the ones obtained by inverse analysis, corresponding to the
primary solution (PS) and after applying Levenberg–Marquardt
optimisation (LM), by the full Levenberg–Marquardt optimisation
Table A1
Examples of geometries considered for the cruciform specimen optimisation design.
Geometry R(mm) L
1
/L
2
(L
2
= 15 mm) b(°)
A 1 1.0 0.00
B 3 1.0 0.00
C 3 2.2 0.00
D 3 2.2 9.46
(a) (b)
A B C D
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
21
εε
max
dd
0
0.05
0.1
0.15
0.2
0.25
0.3
0 0.25 0.5 0.75 1
-0.2 -0.15 -0.1 -0.05 0 0.05 0.1
2
ε
1
ε
Fig. A1. Influence of the geometric parameters on the results of the cruciform test along the axes of the specimen: (a)
e
2
/
e
1
vs. d/d
max
and (b)
e
1
vs.
e
2
, for the geometries
indicated in Table A1: for case A to case B, the fillet radius, R, increases; from B to C, the L
1
/L
2
value increases; and from C to D, the opening angle of the arms, b, increases.
Table 11
Average relative difference results (|d
P
|
M
,jd
evMjMand |d
s
|
M
) obtained for the primary
solution (PS), after the Levenberg–Marquardt optimisation (LM), for the full Leven-
berg–Marquardt optimisation (Full-LM) and classical methodology, like experimental
(CM_E) and analytical (CM_A) data sets.
Strategy |d
P
|
M
(%) jd

e
v
M
j
M
(%) |d
s
|
M
(%)
0x0x=0y0y
PS 0.84 18.01 16.06 13.54 1.43
LM 0.79 13.31 7.89 11.20 1.19
Full-LM 6.62 13.11 35.64 28.95 2.25
CM_E 4.07 13.76 21.10 13.31 2.95
CM_A 1.60 14.39 20.86 13.74 3.40
118 P.A. Prates et al. / Computational Materials Science 85 (2014) 102–120

(Full-LM) and by the classical identification methodology, with like
experimental and analytical procedures (CM_E and CM_A, respec-
tively), for the punch displacement of 60 mm. Also, Fig. 19(b) com-
pares the results of the load evolution as a function of the punch
displacement and Fig. 19(c–e) compares the plots 
e
v
M
vs. d, along
the 0x,0x=0yand 0yaxes, respectively. The relative difference re-
sults of the load (d
P
) and flange draw-in (d
s
¼ðss
ref
Þ=s
ref
) were
determined for all identification procedures, relatively to the mate-
rial under study. Table 11 shows the d
M
P
and jd

e
v
M
j
M
results and the
average absolute relative difference of the sliding,
jd
s
j
M
¼ð1=mÞP
m
i¼1
jd
s
j
i
, between the experimental results and the
ones obtained using identification methodologies. The total num-
ber of points in study for the cross tool results are p= 150, for
d
M
P
,q= 63 for jd

e
v
M
j
M
and m= 180, for jd
s
j
M
.
For the cross-shaped cup test results, the main conclusions are
(see the summary of the results in Table 11 and Fig. 19): (i) glob-
ally, the draw-in of the flange, quantified by jd
s
j
M
variable, is better
described by the inverse analysis methodology, whatever the case
‘‘PS’’ or ‘‘LM’’; (ii) both inverse analysis identification strategies, i.e.
the primary solution (PS) and after Levenberg–Marquardt
optimisation (LM) describe more conveniently the load evolution
results than the classical identification strategies and the full
Levenberg–Marquardt optimisation (Full-LM); the worst results
concern the like experimental procedure (CM_E) and the full
Levenberg–Marquardt optimisation (Full-LM); (iii) the equivalent
strain distributions show similar or better performance for both
inverse methodologies than for both classical identification proce-
dures, except for the case of jd

e
v
M
j
M
(0x), for the ‘‘PS’’ case, and for
the jd

e
v
M
j
M
(0x=0y) and jd

e
v
M
j
M
(0y) values, in the ‘‘Full-LM’’ case;
(iv) concerning the comparison between the two inverse analyses
procedures, the relative difference results jd
P
j
M
and jd
s
j
M
show bet-
ter performance after Levenberg–Marquardt optimisation; this
better performance is not noticeable with regard to the results of
the cruciform test (see Table 9), which means that identifications
with different results, though with similar accuracy, can lead to a
different degree of accuracy when applied to a specific test (see
e.g. Debruyne et al. [20]).
6. Conclusions
This work allowed developing an inverse analysis strategy for
simultaneously determining the constitutive parameters of Hill’48
yield criterion and Swift isotropic work-hardening law, from a sin-
gle test, the biaxial tensile test of a cruciform sample. The sample
geometry has been designed in order to guarantee strain heteroge-
neity, exhibiting strain paths from uniaxial up to equibiaxial
tension. The inverse strategy consists on determining a primary
solution, based on an optimisation methodology that compares
the experimental and numerical results, the latter obtained by
making variations of the constitutive parameters from an initial
solution, according to an algorithm previously built up from a
irect analysis study. Optionally, a gradient-based algorithm, Leven-
berg–Marquardt method can be used, for enhancing the primary
solution. The proposed identification approach leading to the pri-
mary solution was shown to be competitive with classical strate-
gies. The classical strategies make use of a large number of linear
strain-path tests and are time consuming and expensive, as well
as require complex and sensitive analysis of the results. The pro-
posed inverse analysis strategy only requires the measuring of
the load evolutions during the biaxial tensile test of the cruciform
specimen and the evaluations of the equivalent strain distribution
along the axes of the specimen, at a given moment of the test. This
simplicity coupled with the wide range of strain paths allowed by
the design of experiments, and also the circumstance that is not
mandatory the use of an optimisation algorithm, such as Leven-
berg–Marquardt, represent advantage over identification strate-
gies previously proposed, related to the use of full-field
measurement methods. Additional studies will focus on experi-
mental identifications and on extending the presented methodol-
ogy for identifying parameters of more complex plastic models,
i.e. with high number of parameters to be identified.
Acknowledgements
This research work is sponsored by national funds from the
Portuguese Foundation for Science and Technology (FCT) via the
projects PTDC/EME–TME/113410/2009 and PEst-C/EME/UI0285/
2013 and by FEDER funds through the program COMPETE – Prog-
rama Operacional Factores de Competitividade, under the project
CENTRO -07-0224 -FEDER -002001 (MT4MOBI). One of the
authors, P.A. Prates, was supported by a grant for scientific
research from the Portuguese Foundation for Science and Technol-
ogy. All supports are gratefully acknowledged.
Appendix A. Cruciform specimen geometry optimisation
The cruciform specimen geometry was studied based on the fol-
lowing geometric parameters [28]: (i) fillet radius, R, (ii) L
1
/L
2
ratio
and (iii) the opening angle of the arms, b(see Fig. 1).
The design of experiments was performed for an isotropic
material and a wide set of geometries, which include the four
geometries exemplified in Table A1. The aim was to select: (i)
the value for the L
1
/L
2
ratio in order to cover as much as possible
strain paths that commonly occur in sheet metal forming pro-
cesses, from uniaxial to biaxial tension (L
1
and L
2
define the dimen-
sion of the arms and the dimension of the square central region of
the specimen, respectively); (ii) the value of Rthat maximises the
strain value attained in the centre of the specimen, while minimis-
ing the stress concentration effect in the fillet region; and (iii) the
value of bthat guarantees a smooth gradient of the ratio between
the maximum and minimum principal strains along the arms of
the specimen, so that the relative representation of all strain paths
between uniaxial and biaxial tension is balanced, whilst ensuring a
relatively high strain value in the centre of the specimen. Fig. A1
shows the strain paths, at points placed along the axes of the sam-
ple, for the geometries indicated in Table A1: (i) in Fig. A1(a), the
strain path is defined by the ratio
e
2
=
e
1
as a function of the relative
distance to the centre of the sample, d=d
max
, where d
max
is the dis-
tance of A and B points to the centre of the sample (see Fig. 1); and
(ii) in Fig. A1 (b) the strain path is represented by the
e
1
vs.
e
2
dis-
tributions. The chosen geometry, D(shown in Fig. 1) (i) covers a
relatively wide range of strain paths (from biaxial tension to uniax-
ial tension), with (ii) a relatively high strain value in the centre of
the specimen, and (iii) a relative smooth gradient of the maximum
and minimum principal strains along the arms of the specimen.
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