AN ATTEMPT TO IDENTIFY HILL’S YIELD CRITERION PARAMETERS
USING STRAIN FIELD MEASURED BY DIC TECHNIQUE
H. Haddadi and S. Belhabib
CNRS-LPMTM, Université Paris 13 – Institut Galilée
99, av. J.-B. Clément, 93430 Villetaneuse, France
ABSTRACT
This study in an attempt to identify Hill’s criterion by exploiting strain fields and force-displacement curve. The sample used for
the mechanical test is non-standard, it was developed to ensure a wide heterogeneous strain field covering the gauge area for
the identification of material parameters purpose. This kind of identification method is very attractive; it is targeted to identify
constitutive laws with a minimal number of non-standard mechanical tests. The identification strategy adopted here consists in
the minimization of a cost-function representing a weighted least square difference between the simulated and measured data.
Strain fields were measured by digital image correlation (DIC) technique. The studied material is a Dual-Phase steel used in
automobile industry. First results show that identification using strain fields and force-displacement curve gives acceptable
values of Hill’s criterion parameters.
Introduction
The identification of material parameters is very important in order to perform realistic prediction of the material behaviour by
numerical simulations. An alternative to the classical methods of identification consists in the exploitation of the strain field
measured on a non-standard specimen [1, 2]. We have shown in previous studies the improvement brought by the exploitation
of the strain field to the quality of the identification of the material hardening parameters [3, 4]. The aim of this work is to check
the feasibility of the identification of Hill’s yield criterion parameters using full-field strain measurements and forcedisplacement response of a unique non-standard sample.
Material behaviour
The studied material is assumed to have standard rate independent elasto-plastic behaviour. The elasticity is linear and
isotropic. The yield criterion of plasticity used is the Hill’48 [5]. It is given by:
f (σ ,ε p ) = σ − Y(ε p ) = 0
(1)
where
σ2
= F( σ yy − σ zz )2 + G( σ zz − σ xx )2 + H( σ xx − σ yy )2 + 2Lσ yz2 + 2Mσ zx2 + 2Nσ xy2
σ is the equivalent stress of Hill, σ ij
(2)
denotes the components of the stress tensor of Cauchy in the orthotropic axes and
ε p is the equivalent plastic strain.
The work-hardening is assumed to be isotropic. Its evolution is explicitly given by a Swift law:
Y(ε
p
⎛
εp ⎞
) = Y0 ⎜1 + ⎟
⎜
ε 0 ⎟⎠
⎝
n
(3)
The parameters to be identified are F, G, N for Hill’s yield criterion, and Y0 , ε 0 , n for the hardening Swift law. Notice that
G+H=1, this is obtained by imposing the equality between the measured yield limit and the equivalent stress given by Hill’s
yield criterion for a uniaxial traction along the rolling direction. We assume also that L=M=1.5. These laws were implemented in
1
an FE code called EPIM3D [6]. The geometry of the used sample was developed to increase the heterogeneity of the strain
field, ensure divers strain-paths and permit a good sensitivity to the material parameters [3, 4].
Measurement of the in-plane deformation
The used technique for displacement field measurement is the digital image correlation technique. This needs a vision system
in order to acquire images of the deformed surface of the sample during the test. A correlation package is necessary to
process the acquired images and give access to the displacement field. The strain field is obtained by numerical differentiation.
The vision system used in this study is a 4 Mega Pixels CCD sensor providing 8 bit grayscale images. The camera is placed in
front of the sample surface. Fibre light guides are used as lighting source. Given that the sample surface of the studied
material was not textured, a random speckle pattern was deposited on the sample surface using black and white paint. Images
are processed by a graphical user's interface software called "Sept-D" developed by Vacher et al.[7].
Identification of Hill’s parameters
a)
with classical method
Classical method consists in two totally independent identifications. The first one is the identification of the hardening
parameters achieved on a classical tensile test and the second one concerns the identification of Hill’s yield criterion
parameters using seven classical tensile tests.
1-identification of Swift parameters
The identification of Swift parameters was performed using the mean values curves of stress vs. strain obtained during a
classical tensile test along the rolling direction. The identification is based on the minimization of a least square difference,
called also a cost-function, given by:
2
1 exp ⎛ σ Swift ( Ps , ε i ) − σ exp (ε i ) ⎞
⎟ ,
∑⎜
n exp i =1 ⎝
σ exp (ε i )
⎠
n
Fs (Ps ) =
(4)
where σ exp (ε i ) and σ Swift (Ps , ε i ) are respectively the experimental and the Swift model value of stress corresponding to the
strain
εi .
Ps = {Y0 , ε 0 , n} is the parameters set of the Swift model and
The identified values of the parameters are ε 0 = 1.66 × 10 ,
experimental and the Swift model average stress-strain curves.
−3
Y0 = 330
n exp
is the number of the experimental points.
MPa,
n = 0.187
[8,9]. Figure 1 shows the
900
800
700
experience
model
σ [MPa]
600
500
400
300
200
100
0
0
0.05
0.1
0.15
0.2
0.25
0.3
p
ε []
Figure 1. Experimental and identified stress-strain curves obtained from tensile test.
2
2-identification of Hill’s parameters
The coefficients of Hill's criterion were identified using the curve r = r(α ) of Hill's coefficient of anisotropy vs. the angle of the
tensile test loading with respect to the rolling direction. These coefficients were fixed so that the theoretical values of the
anisotropy coefficient of Hill r = r(α ) , given by Equation (6), fit well the experimental values given in Table 1. As for Swift
parameters, the coefficients of Hill’s yield criterion were identified by minimizing a cost-function based on a weighted least
square difference given by:
FH (PH ) =
1
nα
2
⎛ r H (PH ,α j ) − r exp (α j ) ⎞
w (α j ) ⎜
⎟⎟ ,
∑
⎜
r exp (α j )
j =1
⎝
⎠
nα
(5)
with
⎧ 1 if α = 0° or 90°
w(α ) = ⎨
,
⎩2 otherwise
and
r H (PH ,α ) =
where
nα
ε22p 4 − 4G − (F + 4 − 3G − 2N)sin 2 (2α )
=
,
ε33p
2 ( (G − F) cos 2α + F + G )
(6)
is the number of the angles of the tensile tests, r H (PH ,α j ) and r exp (α j ) are, respectively, the theoretical and the
experimental values of the anisotropy coefficient of Hill at the jth angle.
criterion.
α (°)
r = r(α )
PH = {F,G, N} is the parameters set of Hill's yield
0
15
30
45
60
75
90
1.01± 0.03
0.90± 0.01
0.82± 0.01
0.76± 0.04
0.88± 0.01
0.98 ± 0.02
0.98 ± 0.07
Table 1. Mean values of Hill’s coefficient of anisotropy and the related deviation for different
directions of tensile test axis with respect to the rolling direction.
For each value of the angle α three tests were performed. Table 1 shows the average value and dispersion of Hill's coefficient
of anisotropy. The identified coefficients of Hill's yield criterion are: F=0.490, G=0.504 and N=1.270 . The evolution of the
experimental and modelled Hill’s coefficient of anisotropy vs. orientation of the tensile test axis with respect to the rolling
direction is depicted in Figure 3.
b) by exploiting the strain field
We present here the proposed method for the identification of material parameters, its implementation and the obtained
results. Figure 2 depicts the identification block diagram, where the simulation is started with an initial parameters set. After
computing the cost-function, the convergence conditions are checked, if not satisfied a new parameters set is generated and
the cost-function is recomputed again. This procedure is repeated until meeting the convergence conditions. The optimization
©
procedure is based on Matlab routines which use the simplex algorithm [10]. Notice that the boundary conditions applied to
the simulation are measured experimentally; in this study mean values of the measured displacement are applied.
3
Figure 2. Block diagram of the identification strategy
The identification strategy consists in the minimization of the overall cost-function Φ given by the following expression:
With
Φ ( P ) = (1 − β1 − β 2 )Φ F (P ) + β1Φ ε (P ) + β 2Φ r (P )
(7)
Φε (P ) = λ1Φε xx (P ) + (1 − λ1 )Φε yy (P )
(8)
2
sim
exp
1 n F ⎛ F (t i , P) − F (t i ) ⎞
Φ F (P) = ∑ ⎜
⎟ ,
exp
⎟
n F i =1 ⎜⎝
F (t i )
⎠
Φε xx (P ) =
Where Φ F , Φ ε
1
n img
(9)
2
n
sim
exp
1 mp ⎛ ε xx ( X j , t i , P) − ε xx ( X j , t i ) ⎞
⎜⎜
⎟⎟ ,
∑
∑
exp
ε xx ( X j , t i )
i =1 n mp j =1 ⎝
⎠
n img
(10)
and Φ r represent a least square difference between the measured and the simulated total force, strain field
and Hill’s coefficient of anisotropy r = r(α ) respectively. β1 and β 2 are weighting coefficients of the overall cost-function, so
that β1 + β 2 < 1 . λ1 is the weighting coefficient of the strain field cost-function, so that λ1 ≤ 1 . n img is the number of fields and
n mp is the number of virtual grid nodes per field. P = {F,G,N, Y0 , ε 0 , n} is the parameters set to identify.
4
Results and discussion
The obtained results are given in table 2. The identifications identif.#1, identif.#2 and identif.#3 denote, respectively, the
classical identification method, the identification using total force curve and strain fields, identification using total force curve,
strain fields and r = r(α ) curve.
1.5
experience
identif. #1
identif. #2
identif. #3
r[]
1
0.5
0
0
10
20
30
40
50
α [°]
60
70
80
90
Figure 3. Variation of Hill’s coefficient of anisotropy vs. orientation
of the tensile test axis with respect to the rolling direction. Identif.#1: classical identification method; Identif.#2: identification
using total force curve and strain fields; Identif.#3: identification using total force curve, strain fields and r = r(α ) curve.
Figure 4. Experimental and identified longitudinal Green-Lagrange strain field ε xx at U=3.95 mm : (a) experience,
(b) classical identification method; (c) identification using total force curve and strain fields ( β1 = 0.005, β 2 = 0 and λ1 = 0.95 );
(d) identification using total force curve, strain fields and r = r(α ) curve ; ( β1 = 0.0100, β 2 = 0.5 and λ1 = 0.95 ).
5
Figure 5. Experimental and identified transversal Green-Lagrange strain field ε yy at U = 3.95 mm : (a) experience,
(b) classical identification method; (c) identification using total force curve and strain fields ( β1 = 0.005, β 2 = 0 and λ1 = 0.95 );
(d) identification using total force curve, strain fields and r = r(α ) curve ; ( β1 = 0.01, β 2 = 0.50 and λ1 = 0.95 ).
Identif.#1
Weighting coefficients of Φ
Weighting coefficients of Φε
Swift’s parameters
Hill’s parameters
Cost-functions
Identif.#2
Identif.#3
Φε
β1
-
0,005
0.01
Φr
β2
-
0
0.50
ΦF
1 − β1 − β 2
-
0.995
0.49
Φε xx
λ1
-
0.95
0,95
Φε yy
1 − λ1
-
0.05
0,05
Y0 (MPa)
330.3
301.2
296.3
ε0
1.66 x10-3
2.97 x10-3
2.47 x10-3
n
0.187
0.236
0.229
F
0.4900
0.5534
0.4878
G
0.5040
0.5250
0.5059
N
1.270
1.105
1.269
ΦF
4.57 x10
Φε
5.24 x10-2
9.00 x10-3
8.49 x10-3
Φr
8.55 x10
-4
4.68 x10-2
8.35 x10-4
Φ
3.20 x10-3
4.00 x10-4
6.82 x10-4
-3
3.57 x10
Table 2. Identified hardening and yield criterion parameters.
6
-4
3.66 x10-4
Figure 3 shows the obtained r = r(α ) curves for the different identifications. Best results are obtained for identification using
classical method and the one using experimental data of the non-standard sample but that takes into account the curve
r = r(α ) coming from standard tests. The identification method that uses only strain fields and F-u curve gives an r = r(α ) curve
having a profile close to the one identified on the experimental values of Hill’s coefficient of anisotropy curve. We should point
out, however, that the obtained results depend relatively on the weighting coefficients. The proposed method of identification
gives good agreement concerning strain fields contrary to the classical method that overestimates them (Figures 4 and 5). The
identification identif.#3 shows that it is possible to obtain good results on strain fields, F-u curve and r = r(α ) curve at the same
time. In order to improve the obtained results concerning the identification of Hill’s parameters using experimental data of a
non-standard test, it seems necessary to add another non-standard test with an angle between the loading direction and the
rolling direction different from 0°.
Conclusions
Hill’s yield criterion and Swift parameters were identified satisfactorily using a unique non-standard tensile sample. This study
shows the feasibility of the identification of both hardening and yield criterion simultaneously. The proposed identification
method is interesting because it permits to reduce the number of mechanical tests needed for the identification. In our case
one non-standard tensile test was used instead of seven classical ones. In addition to this, the method gives good results
concerning the prediction of strain fields and force displacement response. The obtained Hill’s parameters are indeed
acceptable but their identification should be improved in order to reproduce as accurately as possible the experimental
r = r(α ) curve. This is the objective of the next step of this study where the identification of Hill’s yield criterion will be achieved
(i) using a unique more suitable non-standard sample different from the one used here, or (ii) using two non-standard tests
with an angle between the loading direction and the rolling direction different from 0° for one of them.
Acknowledgments
The authors would like to acknowledge Pr. L. F. Menezes and co-authors from the University of Coimbra for providing an open
source version of the finite element code Epim3D. They also would like to thank Pr. P. Vacher and co-authors from the
University of Savoie for providing free of charge licences of the image correlation software "Sept-D" to LPMTM Laboratory.
References
1.
Mahnken, R. and Stein, E., “A unified approach for parameter identification of inelastic material models in the frame of the
finite element method”. Comput. Methods Appl. Mech. Engr, 136, 225-258, 1996.
2. Meuwissen, M. H. H., Oomens, C. W. J., Baaijens, F. P. T., Petterson, R., and Janssen, J. D., “Determination of the
elasto-plastic properties of aluminium using a mixed numerical-experimental method”. J. Mater. Process. Tech., 75(13):204.211, 1998.
3. Belhabib, S., Haddadi, H., Gaspérini, M., and Bouvier, S., “Mesures par corrélation d’images et simulations numériques
du champ de déformation au cours d’un essai de traction hétérogène”. In Congrès Français de Mécanique, Troyes, 2005.
4. Haddadi, H., Belhabib, S. Gaspérini, M. and Vacher, P., “Elasto-plastic constitutive laws parameters identification using
th
strain field measurements” In Proc. 8 European Mechanics of Materials Conference, Edited by O. Allix, Y. Berthaud, F.
Hild and G. Maier, Cachan – France, 2005.
5. Hill., R., “A theory of yielding and plastic flow of anisotropic metals”. Proc. Roy. Soc., 193(1033):281-297, 1948.
6. Menezes., L. F., “Modélisation tridimensionnelle et simulation numérique des processus de mise en forme : application à
l’emboutissage des tôles métalliques” ,PhD thesis, University of Coimbra - University Paris 13, 1994.
7. Vacher, P., Dumoulin, S., Morestin, F., and Mguil-Touchal. S., “Bidimensional deformation measurement using digital
images”. Proc. Instn. Mech. Engrs., 213 Part C ImechE:811.817, 1999.
8. Haddadi, H., Bouvier, S., and Levée, P., “Identification of a microstructural model for steels subjected to large tensile
and/or shear deformations”. J. Phys. IV, 11:329-337, 2001.
9. Haddadi, H., Bouvier, S., Banu, M., Maier, C., and Teodosiu, C., “Towards an accurate description of the anisotropic
behaviour of sheet metals under large plastic deformations: Modelling, numerical analysis and identification”. Int. J.
Plasticity, 22(12):2226.2271, 2006.
10. Lagarias, J. C., Reeds, J. A., Wright, M. H., and Wright, P. E., “Convergence properties of the nelder-mead simplex
method in low dimensions”. SIAM Journal of Optimization, 9(1):112–147, 1998.
7