IDENTIFICATION OF STRAIN-RATE SENSITIVITY PARAMETERS OF
STEELS WITH AN INVERSE METHOD
Lorenzo Peroni, Marco Peroni, Giovanni Belingardi
Dipartimento di Meccanica, Politecnico di Torino
Corso Duca degli Abruzzi 24, 10129 Torino, Italy
ABSTRACT
Since the second half of the past century, research in the automotive industry focused on vehicle performance, environmental
compatibility, and safety improvement. Increasing interest about passive safety and the ever stricter regulations, both in the US
and EU, pushed towards more refined models of the vehicle behavior and, to achieve this, to a deeper insight into material
behavior. For what concerns crash simulations, one of the most critical uncertainty is related to the influence of the loading
velocity on the material mechanical properties. This influence can be accounted by means of the strain-rate sensitivity that is
substantial for most materials. Strain-rate sensitivity of low-carbon steels, such as the deep-drawing steels of most automotive
body, is very well known.
Aim of the present work is to show a combined experimental and numerical technique, based on a inverse approach, to
determine strain-rate sensitivity parameters of steels. This technique is based on the numerical simulations, by means of a
finite element explicit code, of experimental tensile tests. The tests are conducted at different speeds, from quasi-static loading
conditions with a general purpose hydraulic testing machine, to high speed with a Hopkinson Tensile Bar. This latter device is
-1
able to induce strain-rates of the order of magnitude of some thousand of s . Each test, independently by the used test rig,
substantially gives a load-stroke curve.
Usually stress-strain relations are obtained, analytically, from these experimental data: however the particular geometry and
small dimension of the specimen used in dynamic tensile tests introduces some error essentially due to the triaxiality of the
stress and strain fields. To solve this problem, and improve the accuracy of the results, a series of simulations (LS-Dyna) in the
same conditions of experimental tests are performed, changing material strain and strain-rate law parameters in each run by
means of an optimization strategy (Hyperstudy).
The parameters that give the minimum error, i.e. give the best fit of the numerical result to the experimental one, are the
identified parameters for the material.
Introduction
The Split Hopkinson Pressure Bar (SHPB) has become today a very popular experimental technique for the study of the
constitutive laws of the materials at high strain rates. Historically, the first use of a long thin bar to measure the pulse shape
induced by an impact is considered due to Hopkinson [23]. This method has been well established after the critical work of
Davies [10]. The experimental setup with two long bars and a short specimen has been introduced by Kolsky [26,27].
The split Hopkinson bar technique, which has been initially used in compression, has been extended to tension [21] and
torsion [14].
Kolsky’s original SHPB analysis is based on some basic assumptions. (i) The waves that propagate in the bars can be
described by the one-dimensional wave propagation theory. (ii) The stress and strain fields in the specimen are uniform in the
axial direction. (iii) The specimen inertia effect and the friction effect in the compression test are negligible. Those assumptions
have been extensively studied in past decades. Following Davies works [10], a more accurate wave propagation theory has
been used in the data processing. The oscillations due to wave dispersion effects observed in the average stress–strain curve
have been diminished [17,20,21,29,38].
The assumption of axial uniformity of stress and strain fields permits to relate the average stress–strain curve to forces and
velocities measured at both faces of specimens. Investigations on this point have been reported by Conn [8], Hauser [22] and
Jahsman [25], using a one-dimensional simulation of the wave propagation in the specimen. A two-dimensional numerical
simulation is given by Bertholf and Karnes [4]. Experimental observations of the strain field using the diffraction grating
technique have been reported by Bell [3]. It has been proved then that stresses and strains are not axially uniform, especially
in the early stage of the test.
A typical SHPB setup is outlined in Fig. 1. It is composed of long input and output bars with a short specimen placed between
them. The impact of the projectile at the free end of the input bar develops a compressive longitudinal incident wave. Once this
wave reaches the bar specimen interface, a part of it is reflected, whereas another part goes through the specimen and
develops in the output bar the transmitted wave. Those three basic waves recorded by the strain gages bonded on the input
and output bars allow for the measurement of forces and velocities at the two faces of the specimen.
Output bar
Input bar
Output bar
Input bar
Striker
Striker
Figure 1. Split Hopkinson Pressure Bar in Reliability and Safety Laboratory of Politecnico di Torino (Vercelli Campus)
This measurement technique is based on the wave propagation theory and on the superposition principle. According to the
wave propagation theory, the stress and the particle velocity associated with a single wave can be calculated from the
associated strain measured by the strain gages. Using the superposition principle in an elastic bar, the stress and the particle
velocity in one section are calculated from the two waves that propagate in opposite directions in this section.
As the three waves are not measured at bar–specimen interfaces in order to avoid their superposition, they have to be shifted
back from the position of the strain gages to the specimen faces, in time and distance. This shift leads to two main
perturbations. First, waves change in their shapes on propagating along the bar. Second, it is very difficult to find an exact
delay in the time shift to ensure that the beginnings of the three waves correspond to the same instant. Those perturbations, if
not controlled, can introduce errors in the final result.
By the use of a suitable collar that protects the specimen during the passage of the first compression wave [41], also tensile
tests can be performed; in fact when the compression wave reaches the output bar free end, it is reflected back through the
output bar itself as a tensile wave that travels toward the specimen and now it constitutes the incident wave that will be
partially reflected and partially transmitted by the specimen.
In this cases, the specimen is not a simple cylinder, as in compression tests, but it is necessary to have two screwed
extremities to fix the specimen in the bar, and a reduced section in the middle to avoid fracture at the fixtures (like in quasistatic tensile tests). In this condition the assumption of uniform stress and strain fields in the specimen is not verified.
Numerical simulation
Numerical simulation by means of the finite element methods offers nowadays a convenient and feasible tool to design
structural components even in very complicated environments such as crash loading. Numerical software codes such as
Dyna3D (and its progeny) help in the reduction of time to market and design cost of vehicles, by decreasing the need for real
model crash tests that could be substituted by virtual tests.
Recent development in computer technology allow the use of fine meshes in real industrial numerical models, within a
reasonable calculation cost, so that the accuracy of the description of material behaviors becomes more and more important in
those applications.
It is now also possible perform optimization analysis in non-linear problem with an high number of simulation on the same
problem to evaluate the influence of problem parameters.
For the case where the classical SHPB analyses do not give acceptable results, an identification technique based on an
inverse calculation method is possible. It permits to relate material properties to forces and particle velocities measured at both
faces of the specimen without using the assumption of axial uniformity of stresses and strains [4].
Figure 2. Numerical model of SHPB in tension: detail of specimen and strain signals
For material such a metals and metals alloys, a great number of models have been proposed in past decades but only a few
of them have been effectively used in commercial explicit FEM codes. One reason is that most of the models are quite
complicated and involve a great number of parameters. An other one is that the significance of each parameter is not so clear
and its identification from some simple macroscopic tests is difficult. Models used in numerical simulations are mostly of
phenomenological type, based on macroscopic analysis of experimental data.
The most simple and robust elasto-plastic models with rate-sensitive flow stresses are often used in explicit codes [36]. It is
nevertheless observed that existing models of this type are too simplified and are not suitable over a large range of strain rate.
The precision of the parameter identification is indeed as important as the theoretical quality of the model itself, especially
when the dynamic test data are involved because the loading condition is complex. An identification method of the parameters
of material model from simple experimental data is here presented taking into account the testing condition of dynamic tests
such as the non-homogeneous field of stress, strain and strain-rate distributions in the specimen and the variation of the strainrate during the test.
With the up to date calculation instruments it is quite simple to reproduce a whole Hopkinson bar test in a reasonable
calculations time with results in good agreement with experimental results (fig. 2), reproducing the problem geometry with
accuracy. Of course, it is necessary to correctly model the behaviour of the specimen material.
The rate sensitive elasto-plastic models
Elasto-plastic strain-rate sensitive models such those proposed by Johnson and Cook [37], Zerilli and Armstrong [38] and
Steinberg and Lund [39] are often available in explicit codes. The models of Zerilli and Armstrong or Steinburg are more
specialized in military application in the case of very high strain rates, high temperatures and high hydropressure.
Most used in civil and automotive application is Johnson and Cook model, that gives a linear increase of flow stress with the
logarithm of strain rates:
m
⎞ ⎞ ⎛⎜ ⎛ T − 300 ⎞ ⎞⎟
⎟⎟ ⎟ ⋅ 1 − ⎜⎜
⎟⎟
⎟
⎝ 0 ⎠ ⎠ ⎜⎝ ⎝ Tmelt − 300 ⎠ ⎟⎠
⎛
⎛ ε&
σ = A + Bε p n ⋅ ⎜⎜1 + C ln ⎜⎜
ε&
(
)
⎝
(1)
In their purposes they would include the effects of work-hardening, strain-rate and temperature. In (1) A is the elastic limit
strength, B and n are the work hardening parameter and exponent, C is the strain-rate parameter, Tmelt is the temperature
parameter (corresponding to the melting point of the material) and m is the temperature exponent. ε& 0 is a strain-rate reference
constant that is usually suggested to choose equal to 1 1/s.
In our analysis the temperature influence has not been taken into account since it is not important for the current application.
Also Cowper-Symonds [36] model has been used to consider strain-rate effects. In the original formulation the strain-rate value
only modifies the yield stress and does not influence the slope of the stress-strain curve after yield:
σ 0'
⎛ ε& ⎞
= β =1+ ⎜ ⎟
σ0
⎝D⎠
1/ q
(2)
In formula (2) σ0 is the quasi-static yield-stress of the material, and σ0’ is the yield stress at the strain-rate value ε& . The q and
D parameters are strain rate material properties; they change from material to material.
In modern explicit codes the β factor (Cowper-Symonds) can be used to account for strain-rate effect to static elasto-plastic
models: power-law, piecewise, etc.:
⎡
⎛ ε& ⎞
⎟
⎝D⎠
σ = β ⋅ f (ε ) = ⎢1 + ⎜
⎣⎢
1/ q
⎤
⎥ ⋅ f (ε )
⎦⎥
(3)
Proposed analysis methodology
Static tests
The first step of the material characteristics identification is the evaluation of the static characteristic. A series of specimens is
subjected to tensile test in quasi-static conditions. Some of these specimens have standard proportions and some others have
with reduced length, like those used in the dynamic tests.
A three-dimensional numerical model of the specimens is then prepared (fig. 3): a displacement is imposed on the surface
nodes of specimen edge (screwed part).
Through an optimization procedure (Hyperstudy) the σ−ε characteristic of the material is identified, thus the force-displacement
curve obtained by the experimental tests is well reproduced. The used optimization algorithms are the Sequential response
surface method and together with Method of feasible direction .
R=5
Ø5
M10
Lu
Figure 3. Specimen
This paragraph reports in detail the developed procedure applied to a case of static tensile tests performed by means of a
universal servo-hydraulic testing machine on three classes of different specimens made of mild steel. These specimens differ
from each otheronly for the gage length (in particular A: Lu=50 mm, B: Lu=7mm and C: Lu=5 mm) in order to verify the
influence of the specimen geometry on the material parameters obtained by a usual tensile test. A simple power-law has been
n
adopted (with the form σ=kεp in which σ is the true stress, εp is the effective plastic strain and k and n are the two parameters
of the model that is equivalent to Johnson and Cook model with A=0, C=0, T=300). It guarantees a good likelihood with the
experimental data and an optimum calculation speed in the numerical simulations. A Von Mises collapse surface has been
used for the triaxial stress condition evaluation.
Once performed the experimental tests, the k and n coefficients of the adopted model have been extracted both with classical
method (linear regression in bi-logarithmic coordinates of the plastic strain-stress curve) and with the developed procedure
based on numerical optimization. Table 1 and figure 4, contains the obtained results. The index 1 and 2 (A1,A2 etc.) indicates
a replication of the same test.
Specimen
A1
A2
B1
B2
C1
C2
Table 1. Results of static test analysis
Classical
Optimization
K (MPa)
n
K (MPa)
728.1
0.060
697.0
745.4
0.069
657.0
763.5
0.053
735.0
757.8
0.056
740.0
800.9
0.076
723.0
801.0
0.067
717.0
n
0.048
0.040
0.048
0.060
0.055
0.041
For a more immediate evaluation of the results proposed in figure 4 it is useful the comparison among the force-displacement
curves obtained numerically in the above described two modalities for two specimen typologies that have been analysed.
It can be noted that the proposed procedure based on the optimization technique always gives lower numerical values for K
and n.
The figures 5 on the right side show that the technique for the extraction of the material model parameters by numerical
optimization gives very accurate results for all the three specimen typologies with a satisfying description of even the portion of
mechanical characteristic after necking. On the contrary, the classical techniques of extrapolation are affected by an evaluation
error that increases with the decrease of the analysed specimen gage length. Further, it can be observed that even if with this
classical type of elaboration it is not possible to evaluate the necking phenomenon, the identification of the material
1000
0.1
900
0.09
800
0.08
700
0.07
600
0.06
500
n
k (MPa)
parameters is satisfying only for the specimen with the higher gage length.
optimization
classical
mean optimization
mean classical
400
300
200
0.05
0.04
optimization
classical
mean optimization
mean classical
0.03
0.02
100
0.01
0
A1
A2
B1
B2
C1
0
A1
C2
A2
B1
Specimen
B2
C1
C2
Specimen
Figure 4. Results of static test analysis
Specimens A
Specimens A
20
20
Classic fit
16
14
14
12
10
8
A1
fit A1
A2
fit A2
6
4
2
Opti. fit
18
16
Force (kN)
Force (kN)
18
12
10
A1
fit A1
A2
fit A2
8
6
4
2
b
0
0
1
2
3
4
5
0
6
a
0
1
2
Stroke (mm)
Specimens C
5
6
Specimens C
20
Classic fit
18
16
14
14
12
10
C1
fit C1
C2
fit C2
8
6
4
Opti. fit
18
16
Force (kN)
Force (kN)
4
Stroke (mm)
20
12
C1
fit C1
C2
fit C2
10
8
6
4
2
0
3
2
a
b
0
0.5
1
1.5
Stroke (mm)
2
2.5
3
0
0
0.5
1
1.5
2
2.5
3
Stroke (mm)
Figure 5. Comparison between the Force-stroke characteristic obtained from experimental test and numerical simulation. The
material model parameters of the numerical model are those obtained with classical fit and optimimization.
When the gage length decreases the material strength is overestimated in a more evident manner. It is then immediate that
the developed technique is fundamental for the extraction of more significant parameters of the material when working with
particular specimen geometries that introduce significant geometrical effects.
It is well known as when necking begins during a tensile test, a high localization phenomenon takes place and the material in
the necking region (the only zone where plastic deformation and damage of the material is developing experiences an
increasing triaxiality of the stress state.
The necked region is in effect a mild notch. Necking begins at the point of plastic instability where the increase in strength due
to strain hardening fails to compensate the decrease in cross-sectional area. This occurs at the maximum load or at a true
strain equal to n. After that many fine cavities form in this region, and on progress of the strain increment these grow and
coalesce into a crack. In this last phase a damage model is necessary to reproduce the material behavior.
In the first phase of the necking phenomenon and taking into account the very short length of the specimens (that are
substantially notched specimens) it is reasonable to assume that the problem is governed principally by the geometry of the of
the sample. For this reason the performed numerical analysis (without a damage model for the material) allow to reproduce
accurately also the load-stroke characteristic after the maximum load.
Dynamic tests
The velocity laws to be applied to the specimens extremities are evaluated, in this cases, starting from the experimental results
of the tests with Hopkinson bar.
The split Hopkinson pressure bar arrangement can give very accurate measurements of forces and velocities at both sample
faces if the data processing is carefully performed. There remains the second kind problems of SHPB mentioned in the
introduction, which consist of relating material properties to measured forces and velocities at the two specimen faces. The
classical analysis assumes the axial uniformity of stress and strain fields in the specimen. An average stress strain curve can
be obtained from the usual equations.
The wave dispersion effects on longitudinal elastic waves propagating in cylindrical bars have been studied experimentally by
Davies [10]. On the basis of the longitudinal wave solution for an infinite cylindrical elastic bar given by Pochhammer [33], and
Chree [7], a dispersion correction has been proposed and verified by experimental data. Even though the Pochhammer–Chree
solution is not exact for a finite bar, it is found easily applicable and sufficiently accurate [10].
As mentioned, even for a classical SHPB setup (where measured waves are at most shifted one length of the bar), the onedimensional wave propagation theory has been shown to be not accurate enough. If the wave dispersion effects are not taken
into account, the accuracy of the two strain measurement method becomes rapidly insufficient with the increase of the
propagation distance, as indicated by Lundberg and Henchoz [30]. Consequently, a more accurate propagation theory must
be used, to improve the accuracy of the shifting process, by taking account of so called wave dispersion effects [6,35].
Starting from the signal given by the strain gauges it is then possible to reconstruct the speed profile at the specimen extremity
(fig. 6.a) and the loads exchanged with the bars.
10
0.1
0.09
0.08
0.07
6
INPUT side
OUTPUT side
0.06
4
c
velocity (m*s-1)
8
0.05
mean
0.04
2
0.03
0.02
0
0.01
-2
0
0.02
0.04
0.06
0.08
0.1
0.12
Time (ms)
0.14
0.16
0.18
0.2
0
A1
A2
B1
B2
C1
C2
D1
D2
Specimen
Figure 6. Speed profile during a real SHPB test (a); C parameter of Johnson-Cook model obtained with optimization procedure
At this point, similarly to the static case, a procedure of structural optimization varies the dynamic factor value of the material
model in order to reproduce the interface loads found in the experimental tests. A Johnson and Cook material model is used in
this case: for the basic properties A=0, B=K and n that define the quasi-static stress-strain characteristic, the values obtained
with the initially performed tensile tests have been assumed.
The obtained C parameter is reported in fig. 6.b for different type of specimen (A: Lu=0 mm, B: Lu=1 mm, C: Lu=2 mm, D: Lu=3
mm): a different length of the specimen in a Hopkinson test produce a different strain-rate level during the test. Probably
respect to the Cowper-Symonds (C-S) or other models, the Johnson and Cook (J-C) model is not adequate to cover a wide
-1
range of strain-rate dependence [40]; however for the range of Hopkinson tests (1000-6000 s ) a J-C approximation is
sufficient. Eventually it is possible, to combine these results with quasi-static, and medium strain-rate results to obtain the
parameters of a more accurate model.
It is interesting to note that due to the specimen geometry the specimen stress state is at a certain extent triaxial with the
consequent variation even of the field of strain-rate (fig. 6-7).
In particular figure 7 reports, the strain-rate history during a test at different longitudinal position in the specimen, obtained by
the numerical reconstruction of an experimental test.
Analogously figure 8 reports, the strain-rate history during a test at different radial position in the specimen, obtained by the
numerical reconstruction of the same experimental test.
The particular geometry of the specimen, respect to the SHPB in compression, makes the non uniformity of the strain and
stress field greater. Consequently, during a tensile test by means of the Hopkinson bar the strain rate in the specimen has not
the same value point to point and anyway in any point the strain rate varies during the test, even if the specimen extremities
move with quite constant speed (fig. 7). The identification of the strain-rate sensitivity parameters becomes extremely difficult
so it is necessary to consider an inverse identification technique in order to reduce the errors.
8000
6000
5000
4000
3000
2000
a
b
c
d
e
f
g
4000
strain-rate (s-1)
strain-rate (s-1)
6000
c
a
0
-2000
0
1.3
1.35
1.4
-6000
-3
1.45
b
f
strain-rate zz
strain-rate xx
strain-rate yy
-4000
1.25
e
2000
1000
-1000
1.2
d
T=1.27 ms
7000
-2
abcdefg
-1
Time (ms)
g
0
1
2
3
z coordinate (mm)
Figure 7. Distribution of strain-rate in axial direction along the length of the specimen
8000
strain-rate (s-1)
6000
5000
4000
3000
4000
0
1
2
3
4
5
3000
2000
1
2
3
-1000
1.35
Time (ms)
1.4
1.45
1.5
-4000
strain-rate zz
strain-rate xx
strain-rate yy
T=1.27 ms
543210
1.3
5
0
-3000
0
1.25
4
1000
-2000
1000
-1000
1.2
0
2000
strain-rate (s-1)
7000
0
0.5
1
1.5
2
2.5
r coordinate (mm)
Figure 8. Distribution of strain-rate in axial direction along the radius of the specimen
Conclusions
In the present work a combined experimental and numerical technique, based on a inverse approach devoted to determine
strain-rate sensitivity parameters of steels is shown. This technique is based on the numerical simulations, by means of a finite
element explicit code, of experimental tensile tests. The experimental tests are conducted at different speeds, from quasi-static
loading conditions with a general purpose hydraulic testing machine, to high speed, with a Hopkinson Tensile Bar. This latter
-1
device is able to induce strain-rates of the order of magnitude of some thousand of s . Each test, independently by the used
test rig, substantially gives a load-stroke curve.
Usually stress-strain relations are obtained, analytically, from these experimental data: however the particular geometry and
small dimension of the specimen used in dynamic tensile tests introduces some error essentially due to the triaxiality of the
stress and strain fields. To overcome this problem, and improve the correctness of the results, a series of simulations (LSDyna) in the same conditions of experimental tests are performed, changing material strain and strain-rate law parameters in
each run by means of an optimization strategy (Hyperstudy).
The parameters that give the minimum error, i.e. giving the best fit of the numerical result to the experimental one, are the
identified parameters for the material.
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