Proceedings of the SEM Annual Conference
June 1-4, 2009 Albuquerque New Mexico USA
©2009 Society for Experimental Mechanics Inc.
Application of SEĖ method on a multiphase steel
F. LAURO, D. MORIN, G. HAUGOU, B. BENNANI
Laboratory of Industrial and Human Automation Control, Mechanical Engineering and
Computer Sciences, LAMIH – UMR CNRS 8530, University of Valenciennes, Le Mont Houy,
59313 Valenciennes Cedex 9, France.
Corresponding author: [email protected]
ABSTRACT
The need to model fracture in crashworthiness by means of finite element codes is a real challenge for research.
Before implementing fracture criteria, an excellent knowledge of the stresses and strains states in the material just
before the crack appearance is the first condition necessary to insure the model development. At present, most of
the material behaviour laws, for example for steel, are only defined until the maximum force when the necking
occurs. In this paper, the authors propose to use the heterogeneity of the displacement field on the surface of the
tensile specimen as an initial condition to identify behaviour laws. The method developed uses the information in
all the surface zone of the specimen by using a correlation image technique. Stresses, strains and strain rates are
then obtained to build a surface behaviour call the SEĖ surface. By cutting it, the experimental behaviour laws for
a large strains and strain rates range are then defined for model identification. This new technique is applied on a
multiphase steel and highlights the lack of accuracy of the material models for high strains and strain rates.
I)
The SEĖ method
Classical experimental frameworks used to observe and identify material behaviour for dynamic applications
suffer from two main defaults:
the dynamic tensile machines (dynamic actuators, Hopkinson bars, …) are not able to perform tests at
constant strain rate. The behaviour laws of strain sensitive materials deduced are therefore determined
for mean plastic strain rate.
Classical measurement techniques (strain gages, extensometers …) are not able to measure non
homogeneous and/or out-plane displacement fields occurring in necking zone.
Numerical predictions of global variables such as forces and displacements are not affected by these
approximations. However, the local variables such as stress and strain fields are inaccurately computed for
sensitive strain rate materials and/or materials with early presence of necking. Consequently, this will lead to an
inability to handle damage and fracture phenomena accurately.
Nevertheless, the determination of the true stress-strain curve becomes essential when large plastic deformation
and large plastic strain rate are considered in particular to perform ductile damage or fracture analysis. The first
difficulty of this determination is the heterogeneity of the plastic strain field and in the stress field in the crosssection of the specimen after the occurrence of necking. In this case, the stress could be determined by
where ΔF is the force increment on the tensile specimen and ΔS the section variation associated to.
Until now, these values have been difficult to obtain during a tensile test and this has lead to the calculation of an
average stress. To improve the results, Bridgman [1] uses a correction technique by taking the radial and hoop
stresses into account in the calculation of the nominal stress for a round bar specimen. This technique is difficult
to extend to rectangular cross-section specimens because of the stress state which cannot be calculated easily
with a non axisymetric specimen. Consequently, Ling [2] proposed a weighted-average method for determining
uniaxial true tensile stress versus strain relation after necking for strip shaped samples. The true stress-strain
function after the onset of necking is corrected by the weighted average function which is determined by using
lower and upper bounds of the true stress-strain curves. The parameters of this function are identified by means
of an optimisation process by correlating the force displacement curves obtained by the simulation of a tensile test
and its experimental one. This technique was also used for damage models identification [3] but the unique
solution depends on the optimisation algorithm and also the number of experimental data to converge to it.
Based on the same approach as Bridgman, Zhang [4] proposes an extensive three dimensional numerical study
on the diffuse necking with tensile specimen with a rectangular cross-section. An approximate relation is
established between the area reduction of the minimum cross-section and the measured thickness reduction. A
method then is proposed to determine the true stress-logarithmic strain relation from the load thickness reduction
curve. This technique was also extended to anisotropic material by observing the width reduction of the tensile
specimen in addition [5]. Finally, the latest technique was developed by G’sell [6] for the purpose of polymer
materials. An experimental set up dedicated to the analysis of the displacement fields on a tensile test was built to
take the volume variation into account. The damage process in polymer material is so important that it modifies
the volume of the tensile specimen and in this way the total strain. This cannot be introduced in the correction
technique based on the Bridgman hypothesis. G’sell then used a specific experimental technique coupled with the
calculation of the plastic strain in the three principal directions. The displacement in the direction of the tension is
used to control in real time the speed of the machine. For that, a tensile test with a seven spots specimen is
undertaken and the spots displacements are followed by a camera. (figure 1)
Figure 1: Configuration of the seven markers in the video-controlled tensile testing system [6].
The axial strain is then calculated by a Lagrangian formalism between each spot in the axis AE and interpolated
by a polynomial equation along the entire specimen to become the true strain. The transverse strain is calculated
by using the same approach with the spots placed on the horizontal axis FG. Finally, the strain in the thickness is
obtained by an incompressible or transverse isotropic hypothesis. The true stress can then be calculated by using
the same hypothesis and the volume variation simply obtained by (ε11+ε22+ε33). This technique is very interesting
and well adapted for polymers under static loading. As the true strain is calculated during the tensile test in real
time, the speed of the tensile grips is controlled all along the test to keep a constant strain rate which will
determine the true stress-strain curves. Nevertheless, this technique could not be used for higher strain rates
such as in crash simulations due to the limitation of the real time controlling.
So, an original experimental framework has been developed to solve the previous lack of experimental
displacement measurement by using digital image correlation technique on upsetting tensile tests of plate
specimens [7].
The Lagrangian in-plane strain tensor (ε) is deduced from the displacement fields of the ZOI. The heterogeneity of
the strain field in the specimen (ROI) during the upsetting tensile test can be handled (figure 2). The strain field at
the fracture initiation and propagation can be computed if the frame per second of the video enables one to
observe the fracture phenomena.
Figure 2: a) Region and zone of interest (ROI/ZOI) b) Tensile plastic strain field in a tensile test of flat specimen.
An hypothesis on the material behaviour (incompressibility, transverse isotropy …) has to be made to calculate
the through-thickness strain ε33 in order to deduce the tensile stress σ11 in each ZOI by using
(3)
in which f is the force through the ZOI transverse section, S 0 is the initial transverse section and ε22, ε33 are the
transverse strains.
For strain rate sensitive material, the equivalent plastic strain rate is calculated by backward finite difference using
(4)
where
is the plastic strain rate in the tensile direction and
time between two pictures.
For each ZOI, the triple point (
,
) is plotted in the space of stress, plastic strain and plastic strain rate to
form the SEĖ (Sigma, Epsilon, Epsilon dot) material behaviour surface (figure 3a). The material behaviour law is
deduced by cutting the material behaviour surface at the desired plastic strain rate (figure 3b).
Figure 3: a) SEĖ surface construction from ROIs and b) material behaviour law at desired constant strain rate
deduced from SEĖ surface.
The results obtained by this cutting are a set of curves defining the behaviour of the material for a large plastic
strain rate range as well as for a large plastic strain range particularly after necking.
II)
Application on a multiphase steel
Configuration of the pre stressed Hopkinson technique
Hopkinson bars equipment using pre stressed input bar is here described in details and calculation tools used to
access to the constitutive laws are based on the motion equation for 1D elastic wave (equation 5):
(5)
The set-up developed by the authors for the tensile tests programme is dedicated to sheet samples with large
ticknesses. The Hopkinson device is constituted by two cylindrical bars made of an high strength steel alloy with
dimensions close to 7.5 m in length and 11 mm in diameter. An efficient alignment of the bars is required due to
the large length of the device, so that the sample is located at the interface of the two bars with good alignment
conditions. The present Hopkinson bars device is calibrated with geometric and physical properties of the bars so
as to calculate engineering plastic strain rate and plastic stress evolutions with respect to equations 6 and 7.
(6)
where,
R(t)
describes the current reflected pulse after crossing the specimen, LS is the gage length of the
specimen, CIB is the velocity of the pulses’ system in the input bar and  S (t) is the current plastic strain rate
observed on the gage length of the specimen.
(7)
where, T(t) describes the current transmitted pulse recorded on the output bar. S S is the initial cross section of the
specimen, S(t) is representative of the engineering stress evolution with time in the specimen. S OB and EOB are
respectively the cross section and the Young’s modulus of the output bar.
With the pre-stressed Hopkinson bars technique [8], the position of the strain-gages bridges placed along the
input bar have been optimised so as to obtain a duration time up to 2.4 ms (thus similar to a 5.6 m long projectile).
The location of the measurement points is closed to the sample with respect to a Lagrangian diagramme (figure 4)
in order to check the equilibrium of the input and output forces recorded during the test.
L ag ran g ian d iag ram u sin g p re-stressed H o p k in so n b ars
0
1000
2000
3000
4000
5000
6000
7000
8000
9000
10000 11000 12000 13000 14000 15000
0
-0 ,5
-1
T im e (m s)
-1 ,5
-2
-2 ,5
-3
-3 ,5
-4
x-position (m m )
Figure 4: Lagrangian x/t diagramme using pre-stressed Hopkinson bars.
The 3 strain-gage bridges are strategically located, and the typical raw signals are presented in figure 5. The three
complementary elastic pulses – shown in the black rectangle – are analysed to establish the constitutive law for
the considered strain rate. Classical well-know relations [9] are used to determine current displacements and
velocities of each bars in function of time as well as input and output forces.
Figure 5: Typical signals collected after a tensile test with H-shape specimens.
The true plastic strain vs. stress relations are finally established and validated up to striction for tensile tests. The
equations 9 and 10 are mentionned after a correction of the apparent modulus process related in equation 8.
(8)
where CORRECTED(t) is the corrected strain,
desired and apparent elastic modulii.
S(t)
is the engeneering stress, ES and EAPPARENT are respectively the
(9)
where
TRUE(t)
and
S(t)
are respectively true and engineering strains
(10)
where
TRUE(t)
and
S(t)
are respectively true and engineering stresses,
is the Poisson’coefficient
Description of the tensile tests programme
Two sets of 15 H-shape samples have been extracted from a Multiphase steel sheet using electro erosion cutting
with tolerances close to 0.05 mm and refined in the gage area with fine roughness so as to reduce the influence of
the machining conditions on the material responses (figure 6). A first set of samples has gage length with 5 mm
and a second set has gage length close to 10 mm so as to quantify the influence of reduced dimensions on the
local and global plastic strains. The thickness of the samples is close to 1.7 mm with negligible standard deviation.
Before being tested gaps at the interfaces set up/sample have been cancelled using a spring at the end of the
output bar. The loading of the sample is provided after the break of a brittle fuse placed in a locking mechanism
used to lock and pre load the input bar at a required tensile force.
Figure 6: Dimensions of the samples (in mm).
As non contact measurement tools are required, the set of samples have been prepared with a grey scale
speckle. An high speed camera (Photron APX RS 3000) has been placed at 90° of the surface of the sample so
as to ensure the acquisition of the pictures (Figure. 7). All the sequences have been recorded respectively at
37 500 f/s (512 128 pixels) for the first set of samples and 45 000 f/s for the second set of samples with a format
close to 384 128 pixels. A Vishay Ampli-conditioning system with a set of three 350 full-bridges ensure the
signals conditioning of the elastic waves initiated after the release of the elastic energy accumulated in the prestressed part of the input bar. The signals of the elastic waves have been recording using an Yokogawa transient
recorder with sampling rate up to 1 MHz.
Figure 7: Configuration of the tensile test programme – tensile set-up before and after the test.
Presentation of the global material results on tensile tests on Multiphase steel
The mean behaviour laws are presented in figures 8 and 9 for the two set of referred samples (gage length: 5 and
10 mm). The results have been obtained on the basis of the calculations of a representative material response per
strain rate with a set of 2 to 4 tests. The associated strain rates evolutions are then presented so as to illustrate
-1
that the loading conditions can be supposed to be at a quasi constant rate of strain up to 400 s . The present
results have been validated after checking of the equilibrium of the input and output forces.
Figure 8: Synthesis of the tests on specimens (gage length : 10 mm) – Associated range of strain rates.
Figure 9: Synthesis of the tests on specimens (gage length : 5 mm) – Associated range of strain rates.
Presentation of the local material results on tensile tests on Multiphase steel
The authors have exploited the digital image correlation software (Corelli) developed by Hild [7]. Before being
tested a greyscale pattern has been spread on each surface of specimen, and has been validated previously so
as to provide a cohesive film in accordance with large strain as well as uncertainty aspect (out of plane
displacements). As described in the SEĖ method, better results are obtained on heterogeneous tests in order to
access to the largest behaviour surface. Under tensile loadings, strain field heterogeneities are susceptible to
come from micro-structure aspects as well as necking. Figure 10 illustrates local calculations obtained on the
basis of Digital Image Correlation, it is clearly shown that different plastic strain localizations may occur. In
remark, necking appears on a particular site defined by current micro structures conditions.
Figure 10: Longitudinal Lagrangian strain fields
Firstly, figure 11 shows in details strain fields calculated on each region of interest in function with time. As
observed here, local strain fields revealed being heterogeneous before necking appears. After necking, very high
strains are located in the necking area with values strongly higher than global measurements. Outside this
domain, local plastic strains include global plastic strains from classical Hopkinson measurement technique.
Figure 11: comparison between global and local plastic strains and strain rates fields versus time
-4
In the example shown in figure 11, the necking is visible on the specimen up to 5e s. In order to build a behavior
surface, it is required to access to large plastic strains and strain rates, where the first derivative of the strain rate
is still strictly positive during the test.
Finally, the behaviour surface (figure 12a and 12b) of the multiphase steel has been established using a set of 10
specimens of each gage length. As a consequence, a set of true plastic strain/stress laws can be obtained at a
constant true plastic strain rate extracted from a defined surface cutting as shown in figure 13.
Figure 12: a) behaviour surface in strain/strain rate plane - b) 3D view of the surface
Figure 13: True behaviour laws at three constant plastic strain rates
These true behavior laws at constant strain rates give the opportunity to determine a significant hardening
evolution after necking as illustrated with the well-known true strain/stress relations. Secondly, the authors have
-1
observed a saturation effect on the visco plasticity up to 2000 s .
Discussion
Figure 14 illustrate the comparison between a local and a global behaviour law, the first one has been obtained at
a constant strain rate and the second one considers the loadings. In these conditions, a good correlation is
obtained up to necking (0.10 plastic strain) which confirm the validity of the SEĖ method.
=
Figure 14: Comparison between local and global behaviour at constant strain rate
Commercial finite element codes require a tabulated behaviour law or a mathematical model. For visco plastic
models, parameters depend on the material and need to be identified on a large range of strain rates. Parameters
of classical models like Johnson-Cook model are determined below the necking and for small strains (0.06 plastic
strain for example) on global measurements. Figure 15a shows a typically identification of a Johnson Cook plastic
behaviour model combined with a Cowper-Symmonds for visco plasticity.
Figure 15: a) Classical model identification b) Extension of chosen model to large strains and strain rates
The availability of this phenomenological model can be observed on figure 15b. It is noticed that mathematical
response is in good agreement with plastic strains below 0.15, and is not supported for large strains. In these
conditions, future models arranging with large strain and strain rates are needed in accordance with impact and
crashes of transportation structures conditions where failure is reached after large deformation.
III)
Conclusion
In this paper a new methodology to determine behaviour laws for strain rate dependant materials for a large range
of plastic strains and plastic strain rates is proposed. This methodology uses the heterogeneity of the
displacement field observed on the surface of a tensile specimen. This displacement field is measured by a
correlation image technique on different zones on the specimen and the true strains associated are calculated by
a Lagrangian formulism. The corresponding strain rates are obtained from the previous calculations and finally the
true stresses are obtained by using the incompressible or transverse isotropic hypothesis. All this information is
transferred to a behaviour surface called the SEĖ surface. It represents the behaviour of the material in the space
of stress, true strain and strain rate. A set of true stress-true strain laws at constant strain rates are then obtained
by cutting the SEĖ surface by the plane area defined by constant strain rates. Behaviour models could then be
identified by means of this set of curves.
This new technique has been applied to a multiphase steel, the behaviour laws obtained before necking are in
good correlation with ones obtained by classical techniques. Moreover, after necking enough information are
available with this new technique to perform the identification of materials models for high strains and high strain
rates based on experimental data. In consequence, materials models which are defined before necking could be
not enough accurate for this range of high strains. This work highlights the great interest of the SEE method.
Acknowledgments
The present research work has been supported by International Campus on Safety and Intermodality in
Transportation, the Nord-Pas-de-Calais Region, the European Community, the Region Delegation for Research
and Technology, the Ministry of Higher Education and Research, and the National Center for Scientific Research.
The authors gratefully acknowledge the support of these institutions.
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