Strojniški vestnik - Journal of Mechanical Engineering 62(2016)4, 220-230

Received for review: 2015-12-02

© 2016 Journal of Mechanical Engineering. All rights reserved. Received revised form: 2016-01-28
DOI:10.5545/sv-jme.2015.3266 Original Scientific Paper Accepted for publication: 2016-02-23

Estimating the Strain-Rate-Dependent Parameters

of the Cowper-Symonds and Johnson-Cook Material Models
using Taguchi Arrays
Škrlec, A. – Klemenc, J.
Andrej Škrlec* – Jernej Klemenc
University of Ljubljana, Faculty of Mechanical Engineering, Slovenia

In order to reduce R&D costs, a product’s behaviour during use is predicted with numerical simulations in the early phases of R&D. If a
structure is subjected to high-strain-rate loading this effect should be considered in the material models that are used for the numerical
simulations. This article shows how the strain-rate-dependent material parameters can be determined by combining the experimental data
with the numerical simulations. The presented methodology is based on the application of Taguchi arrays to find the most appropriate values
for the strain-rate-dependent parameters. The presented methodology is applied in a practical case, for which the parameters of the Cowper-
Symonds and Johnson-Cook material models are estimated.
Keywords: strain-rate-dependent material behaviour, finite-element method, explicit dynamics, Taguchi arrays

Highlights
• A procedure for modelling the strain-rate-dependent behaviour of materials is presented.
• The parameters of the Cowper-Symonds and Johnson-Cook material models are estimated on the basis of an impact
experiment.
• A simulation plan for estimating the material parameters is developed with the help of Taguchi orthogonal arrays.

0 INTRODUCTION plastic domain. Investigations of this effect have been

performed by many researchers over the last century,
To reduce the costs of a research and development like Hopkinson, Charpy, Taylor [1], Zerilli, Armstrong,
(R&D) process and optimise the product’s design, Johnson [2], [3], etc. In the last 20 years, strain rate
while ensuring the necessary safety, effectiveness influence on material behaviour is still interesting for
and reliability of the newly developed product, finite- researchers like El-Magd [4], Zhao and Gary [5], Huh
element (FE) simulations of the product’s behaviour et al. [6] to [8], etc.
under real operating conditions are widely applied. When using the explicit dynamic FE code to
To obtain reasonable simulation results the operating simulate extreme loading conditions, such as impact
conditions as well as the product’s geometry and its phenomena, the material models that consider the
material properties must be known. Usually, in the strain-rate dependency of the material’s plastic curve
R&D process the structural loads are assumed on the are commonly applied. The three most commonly
basis of similar products, are obtained by numerical applied material models in researches [9] to [15] that
simulations or are defined by the customer. If we consider the strain-rate effects are: Cowper-Symonds,
assume that the structural loads resulting from the Johnson-Cook, and Zerilli-Armstrong. Since the
operating conditions are known, it is the material
Cowper-Symonds and Johnson-Cook material models
properties that have the greatest influence on the
are simpler than the Zerilli-Armstrong material model,
product’s behaviour for a given geometry. If the
we considered only the former two models in our
structure is subjected to extreme mechanical loading
research. The main difference between them is how
conditions (e.g., impacts during crash tests or different
they account for the strain-rate effects. Consequently,
burst tests) it is of tremendous importance to consider
the number of material parameters that describe the
the strain-rate-dependent material properties when
performing a FE simulation. plastic stress-strain relationship with the strain-rate
It is known from the literature [1] that quasi- effects is different ([16] and [17]):
static loading does not have a significant influence • Yield stress according to the Cowper-Symonds
on the material’s yield stress and the stress-strain material model:
relationship, but this changes if the strain rate
   ε  
1/ P
increases. The increased values of the strain rate  E ⋅E p
σ y = σ 0 + β ⋅ t ⋅ ε eff  ⋅ 1 +    ; (1)
cause an increase in the material’s yield stress and  E − Et    C  
change the material’s stress-strain behaviour in the
220 *Corr. Author’s Address: University of Ljubljana, Faculty of Mechanical Engineering, Aškerčeva 6, 1000 Ljubljana, Slovenia, andrej.skrlec@fs.uni-lj.si
 Strojniški vestnik - Journal of Mechanical Engineering 62(2016)4, 220-230

• Flow stress according to the Johnson-Cook as follows. After the introductory section and the
material model [3]: theoretical background, the experimental arrangement
and the experimental results are presented. The article
σ flow = σ 0 + B ⋅ ( ε effp )  ⋅
n

  continues with a presentation of the results and their

  ε p    T − Troom 
m discussion and ends with a concluding section and a
⋅ 1 + c ⋅ ln    ⋅   , (2) list of references.
  ε0    Tmelt − Troom 
where σ0 is the reference yield stress, E is the 1 THEORETICAL BACKGROUND
material’s elastic modulus, Et is its tangent modulus, β
is the hardening coefficient, ε effp is the effective plastic 1.1 Applied Material Models
strain, ε is the strain rate and ε0 is the reference
strain rate. C and P are the strain-rate parameters of In this article two material models (i.e., the Cowper-
the Cowper-Symonds material model; B, n, c and m Symonds model and a simplified Johnson-Cook
are the strain- and strain-rate-dependent parameters of model) that consider the strain-rate effects on the
the Johnson-Cook material model. The Johnson-Cook material’s behaviour are compared. For each material
material model considers the influence of temperature model three parameters were estimated on the basis of
on the stress-strain behaviour, but not the Cowper- the impact experiment by using the explicit dynamic
Symonds material model. However, the temperature simulations combined with a parameter factorization
effects can be omitted from the Johnson-Cook material according to the Taguchi array.
model if its parameter m is set to zero. The temperature The Cowper-Symonds (C-S) material model with
influences can be omitted from the Johnson-Cook a bilinear characteristic is determined with the
material models if they do not influence the material following parameters ([16] and [17]): elastic modulus
behaviour. The Cowper-Symonds and Johnson-Cook E, Poisson’s number ν, tangent modulus Et, hardening
models are the most commonly applied material coefficient β, material density ρ, and the parameters C
models when performing explicit dynamic simulations and P that describe the dependency of the yield stress
up to moderate strain rates, (i.e., up to ε = 104) ([14] σy on the strain rate ε , see Eq. (1). Therefore, the flow
and [15]), at which the changing temperature does not stress is calculated as follows:
have a significant influence.
σ flow = σ y + Et ⋅ ε effp =
The main problem, linked to the material
parameters that consider the strain-rate effects, is  E ⋅E p    ε  
1/ P

   + Et ⋅ ε eff . (3)
p
that they cannot be simply measured and determined, = σ 0 + β ⋅ t ⋅ ε eff
 ⋅ 1 +
 E − Et    C  
thus they are empirically determined through special
experimental and optimisation processes ([9] to For the C-S material model the strain rate
[13]). For the above-mentioned material models the influences only the yield stress σy. This means that the
parameters that consider the strain-rate dependence plastic curves (the flow stress as a function of strain)
were investigated for many different materials. In the are parallel. The larger the strain rate, the higher the
literature, their typical values for mild steel ([9], [10] flow-stress curve, see Fig. 1a. For high-strain-rate
and [11]), high-yield-strength steel [12], aluminium applications the parameters C and P in Eqs. (1) and
alloys [13], titanium alloys [13], etc. can be found. (3) are usually not estimated from the tensile test, due
Since, on the other hand, no link between the strain- to the limitations of the existing tensile-test equipment
rate parameters and the chemical composition of (maximum strain rates are up to a few hundreds of s-1).
the material was found, these parameters should Together with the tangent Et they were determined on
be identified individually for each material under the basis of the impact experiment.
consideration. Since the temperature effects were neglected in
In this article we will compare the performance our case, the simplified Johnson-Cook (J-C) material
of Cowper-Symonds and Johnson-Cook material model was applied with the parameter m from Eq. (2)
models when applied for simulating the impact of a being equal to 0. This material model is determined
ball on a thin steel plate. The parameters of the two with the following parameters ([16] and [17]): elastic
material models, which cannot be estimated from modulus E, Poisson’s number ν, reference yield stress
the tensile test, were determined from the impact s0, exponent of the flow-stress curve n, scale factor
experiments using the LS-DYNA explicit FE code B for the effective plastic strain, sensitivity c to the
that was combined with a Taguchi array to reduce the logarithm of the strain rate and material density ρ. The
numerical processing effort. The article is structured flow stress is then given by:
Estimating the Strain-Rate-Dependent Parameters of the Cowper-Symonds and Johnson-Cook Material Models using Taguchi Arrays 221
 Strojniški vestnik - Journal of Mechanical Engineering 62(2016)4, 220-230

Fig. 1. Flow stress for a) the Cowper-Symonds material model and b) the Johnson-Cook material model

  ε p   reason for this decision was that the use of Taguchi

σ flow = σ 0 + B ⋅ ( ε effp )  ⋅ 1 + c ⋅ ln    . (4)
n
arrays results in a significant reduction in the number
    ε0   of studied parameter combinations. Their advantage
is that the assigned combinations of parameters are
For the J-C material model the strain rate approximately equally distributed over the search
influences the flow stress σflow in its whole range and space, which is not always the case with, e.g., Latin
the flow-stress curves are not parallel. Their non- hyper-cubes.
linearity depends on the exponent n. The larger the We applied a L81(340) Taguchi array and
strain rate, the higher the flow-stress curve, see Fig. transformed it into the L81(910) orthogonal array using
1b. The three parameters B, n and c in Eq. (4) cannot the linear graph in Fig. 2 ([19] and [20]). This was
be estimated from the tensile test and were determined done because we needed as many levels per parameter
from the impact experiment. as possible and L81(910) best suited this requirement.
1.2 Simulation Plan for Material-Parameter Estimation

For each of the two strain-rate-dependent material

models, three parameters (i.e., Et, C and P for the C-S
material model and B, n and c for the J-C material
model) need to be determined. These parameters
are usually determined using a reversed engineering
approach with the help of numerical simulations that
reproduce the actual experiment ([11], [12] and [18]).
This means that a series of numerical simulations with
different combinations of material parameters are
carried out to establish which combination of material Fig. 2. Linear graph for the transformation of a L81(340) Taguchi
parameters best fits the experimental results. The array into a L81(910) array
problem is that the strain-rate-dependent parameters
can be selected from a domain with a range that spans Using the L81(910) orthogonal array the three
over many orders of magnitude. material parameters (Et, C and P for the C-S material
For this reason it is almost impossible to run model and B, n and c for the J-C material model) are
a full-factorial simulation plan to determine the attributed to the first three columns of the L81(910)
optimal strain-rate-dependent parameters, because array. The levels were chosen from a very wide
the processing time would be prohibitive even in the domain to account for all the possible parameter
case when simulations are run on a supercomputer. To values for different kinds of steels. This makes the
shorten the processing time for estimating the material proposed approach general, although it was tested
parameters it was decided to apply orthogonal for the case of E185 steel. Since the ranges of the
Taguchi arrays for a simulation-plan set-up. The individual parameters span over more than one order
222 Škrlec, A. – Klemenc, J.
 Strojniški vestnik - Journal of Mechanical Engineering 62(2016)4, 220-230

of magnitude, the logarithms of these parameters By applying the Taguchi arrays for the simulation
represent the nine-level factors in the Taguchi array. In plan the number of material-parameter combinations
this manner 81 combinations of the parameter triples that need to be simulated was reduced by a factor of
(Et, C, P) and (B, n, c) were obtained that cover the nine when compared to the full-factorial simulation
whole domains of these parameters, see Fig. 3 for the plan.
parameters of the C-S material model and Fig. 4 for
Table 2. Parameter levels for the J-C material model
the parameters of the J-C material model.
Original parameter levels of parameter B [GPa]
Table 1. Parameters levels for the C-S material model 0.1000 0.1778 0.3162 0.5623 1.0000
1.7783 3.1623 5.6234 10.0000
Original parameter levels of parameter C [ms-1] Original parameter levels of parameter n [-]
0.2154 0.4642 1.0000 2.1544 4.6416 0.001 0.0024 0.0056 0.0133 0.0316
10.0000 21.5443 46.4159 100.0000 0.0750 0.1778 0.4217 1.0000
Original parameter levels of parameter P [-] Original parameter levels of parameter c [-]
1.0000 1.7783 3.1623 5.6234 10.0000 0.001 0.0024 0.0056 0.0133 0.0316
17.7828 31.6228 56.2341 100.0000 0.0750 0.1778 0.4217 1.0000
Original parameter levels of parameter Et [GPa]
0.1000 0.1778 0.3162 0.5623 1.0000
For each combination of the material parameters
1.7783 3.1623 5.6234 10.0000
six numerical simulations were carried out to account
for the six different boundary and initial conditions,
see Table 3. Therefore, when applying the simulation
plan according to the L81(910) Taguchi array 81·6 = 486
numerical simulations were performed for each of the
two material models.
A cost function that measures the deviations
of the experimental and simulation results was
defined so that it measured the difference between
the experimentally determined and simulated data
for the indentation depth H and the position of the
indentation centre Z for the specimen. The averages
of the experimentally determined values for these two
geometrical parameters for two specimen thicknesses
and three different velocities of the ball are listed in
Table 3. The cost function that was used to assess
Fig. 3. Distribution of C-S material-parameter combinations over the goodness-of-fit between the experiments and the
their domains simulations is defined as follows for the C-S and J-C
material models:

 k

 wH ⋅ ∑ ( H exp − H sim ) + 
2

1 i =1
 , (5a)
f ( Et , P, C ) =
k k
2
 +(1 − wH ) ⋅ ∑ ( Z exp − Z sim ) 
 i =1 

 k

 wH ⋅ ∑ ( H exp − H sim ) + 
2

1 i =1
 , (5b)
f ( B , n, c ) =
k k
2
 +(1 − wH ) ⋅ ∑ ( Z exp − Z sim ) 
 i =1 
where k is the number of experimental results with
Fig. 4. Distribution of J-C material-parameter combinations over different boundary and initial conditions (according
their domains to Table 3, k = 6). Hexp and Zexp, are the averaged
Estimating the Strain-Rate-Dependent Parameters of the Cowper-Symonds and Johnson-Cook Material Models using Taguchi Arrays 223
 Strojniški vestnik - Journal of Mechanical Engineering 62(2016)4, 220-230

measured maximum indentation depth and its z The measured engineering stress-strain curves are
coordinate for the specimen. Hsim and Zsim are the presented in Fig. 6a and the resulting average true-
maximum indentation depth and its z coordinate stress–true-strain curve is presented in Fig. 6b. This
for the specimen obtained by simulations. wH was a true-stress–true-strain curve was determined with Eqs.
weighting factor in the two cost functions and was (6) and (7), see Dowling [22].
equal to 0.5 in our case.
ε = ln (1 + ε ) , (6)
2 EXPERIMENTAL ARRANGEMENT σ = σ (1 + ε ) . (7)

2.1 Measurement of a Static Stress-Strain Curve ε and σ were the corresponding average
engineering strain and stress, respectively.
The methodology that was described in Section 1
was applied to characterise the strain-rate-dependent
material behaviour of a mild steel E185. Its static 2.2 Experimental Determination of the Material Behaviour
material properties (elastic modulus, yield stress, at High Strain Rates
ultimate tensile stress) were measured according to
the ASTM E8/E8M standard [21] on Zwick/Roell The main objective was to determine the strain-rate-
Z050 testing equipment. dependent material parameters for simulating the
behaviour of a mild-steel sheet metal that is used
as a shield during a turbine burst test. To identify
the corresponding material parameters we designed
experimental apparatus for shooting a steel ball at a
flat specimen. The experimental arrangement was
based on the ASTM D5420 standard [23], which
describes a test method for measuring the impact
resistance of a flat rigid plastic specimen by means
of a striker impacted by a falling weight. During the
impact between the ball and the flat specimen strains
can be measured on the left-hand side of the specimen
with strain gauges. After the impact test the gross
Fig. 5. Zwick/Roell Z050 test stand and the specimen geometric data, i.e., the indentation depth H and the
position of the indentation centre Z, were measured,
The test stand and the specimen geometry are see Fig. 7.
presented in Fig. 5. A total of 21 specimens were The experimental apparatus was mounted on a
tested. The average yield stress and the ultimate tensile testing machine that was originally built for burst tests
strength were 185 MPa and 350 MPa, respectively. on supercharger structures, see Fig. 7.

Fig. 6. a) Measured engineering stress-strain curves; and b) the resulting true-stress–true-strain curve

224 Škrlec, A. – Klemenc, J.

 Strojniški vestnik - Journal of Mechanical Engineering 62(2016)4, 220-230

Fig. 7. a) An experimental device; b) front view and c) upper view of the experimental specimen

To study the different dynamic behaviours of point to the border is progressive and it is difficult to
steel plates with different thicknesses, balls with a calculate the true strain-rate value at the spot of the
diameter of 12 mm and a weight of 7 g were shot at strain gauges. The simulated peak strain rate at the
the centre of the specimens at an angle of 20° with impact point was 5000 s-1.
different velocities that depended on the specimen
thickness, see Table 3. The initial conditions and the Table 3. Combinations of boundary conditions and results from
plate thicknesses where chosen with regard to the experiments
machine’s limitations and the expected application Position of
Average Measured max.
of the results (thin-shelled supercharger burst shield). Experiment Specimen
ball indentation
the max.
condition thickness indentation
The specimens (see Fig. 7) were metal plates with velocity depth, average, depth, average,
number [mm]
dimensions of 98 mm × 60 mm. Two different sheet- [m/s] Hexp [mm] Zexp [mm]
metal thicknesses were tested: 1 mm and 1.5 mm. The 1 1 103 11.37 34.83
specimens were fixed along the shorter side. The free 2 1 109 12.12 34.89
area of the specimen was 60 mm × 60 mm. The impact 3 1 121 13.07 35.11
velocities were measured on the testing machine just 4 1.5 121 10.38 30.19
before the impact point using photo-sensors. For each 5 1.5 131 11.53 29.60
combination of the steel-plate thickness and the impact 6 1.5 139 12.65 30.49
velocity, three test repetitions were usually performed.
The average and the standard deviation of the From the results in Table 3 we can see that the
indentation depth and the position of the indentation scatter of the experimental results is relatively small,
centre are presented in Table 3. Some of the plate which means that the experimental arrangement was
specimens were also equipped with strain gauges appropriate for our study. Furthermore, we can see
for measuring the strains to obtain the strain rates. that the indentation depth increases with the increasing
Such specimen preparation was time consuming and velocity. The indentation depth at a thickness of 1.5
there were only a few specimens for which the strain mm is smaller than for the 1-mm-thick steel plate.
gauges did not break off during the measurements. Besides, it is clear from Table 3 that for the 1.5-mm-
Nevertheless, some strain measurements were thick specimens, the point with the deepest indentation
successful and an agreement between the measured is approximately 30 mm from the lower side of the
and simulated strain rates for the 1-mm-thick plate specimen, just in the centre of the impact. This is not
and the impact velocity of 109 m/s was as follows: the the case for the 1-mm-thick specimens, for which the
measured peak strain rate was approximately 160 s-1, point with the largest indentation is 5 mm from the
whereas the simulated values at that spot were 200 s-1 centre of the impact. The lack of displacement of the
to 250 s-1. The major causes for the difference of 30 % deepest imprint for the thicker plate is a consequence
were the idealisation of the FE model and the FE mesh of the fact that after the plastification almost all of the
resolution, since the strain-rate decay from the impact kinetic energy of the ball was consumed. This was not
Estimating the Strain-Rate-Dependent Parameters of the Cowper-Symonds and Johnson-Cook Material Models using Taguchi Arrays 225
 Strojniški vestnik - Journal of Mechanical Engineering 62(2016)4, 220-230

the case for the thinner plate and the ball proceeded dimensions of the specimen were recorded for further
in its direction of travel, though causing an extension processing, like in the experiment (Fig. 7).
of the imprint in the vertical direction. This implies The C-S material model was applied using the
different impact dynamics for the specimens with MAT_PLASTIC_KINEMATIC (MAT3) material
different thicknesses and this should be replicated by model from LS-DYNA ([16] and [17]). This material
the numerical simulations if the strain-rate-dependent model is defined with the following parameters, see
material parameters are properly identified. also Eqs. (1) and (3): material density, elastic modulus,
Poisson’s ratio, yield stress σ0, tangent modulus Et and
3 RESULTS AND DISCUSSION the C-S parameters P and C. The parameters Et, P and
C were estimated using the procedure from Section 1
3.1 FE Model for Identification of the Material Parameters and the above-described FE model.
The J-C material model was applied using the
MAT_SIMPLIFIED_ JOHNSON_COOK (MAT98)
The LS-DYNA explicit dynamic FE code was used
material model from LS-DYNA ([16] and [17]).
to identify the material parameters of the C-S and J-C
This material model is defined with the following
material models [16]. The FE model that was used
parameters, see also Eq. (2): material density, elastic
to simulate the ball-impact experiment from Section
modulus, Poisson’s ratio, yield stress σ0, and the J-C
2 is presented in Fig. 8. The steel sheet model had parameters B, n and c. The latter three parameters
5436 four-node and three-node shell finite elements. were estimated using the procedure from Section 1
The mesh density around the impact area was larger and the above-described FE model.
than in the wider region of the specimen model, to The values of the other material parameters were
accurately simulate the indentation. The mesh density fixed in our simulation: the material density was 7850
was chosen to optimise the processing time for a kg/m3, the elastic modulus was 2.1·105 N/mm2, the
reasonable accuracy of the deformation. The rigid ball yield stress was σ0 = 185 MPa (see Fig. 6) and the
was modelled with 448 solid finite elements. In the Poisson’s ratio was ν = 0.3. The reference strain rate
finite-element model, the nodes on both sides of the ε0 for the J-C model was 1 s-1 and was taken as a
thin sheet-metal plate were fixed (Fig. 8). A rigid ball default value from LS-DYNA. This value usually
was shot into the centre of the sheet at an angle of 20° follows from the tensile-test arrangement. However,
with different impact velocities, which are presented its choice does not phenomenologically influence the
in Table 3. results, since the variation of the parameter ε0
monotonically influences the changes of the estimated
parameter c:
C1 ε( 1 )
= ln (02 ) . (1)
C2 ε0

3.2 Results and Discussion

Table 1 and Fig. 3 represent factor levels for the

original domains of the C-S parameters Et, P and
C. Table 2 and Fig. 4 represent factor levels for the
original domains of the J-C parameters B, n and c.
For each of the 81 combinations of the three material
parameters for the C-S and J-C material models six
simulations were carried out, i.e. three different impact
velocities combined with two different specimen
Fig. 8. FE model for a ball-plate impact simulation thicknesses. The FE simulations were carried out
on a numerical server with two Intel Xeon X5670
Between the flat specimen and the rigid ball 2.93-GHz processors, 48 GB of RAM and a Linux
was an AUTO_SURFACE_TO_SURFACE contact operating system. The time spent for one numerical
with the friction coefficient μ = 0.2. This was an simulation on one processor’s core was about 2 hours.
approximate average value from different references The values of the cost function from Eq. (5a)
[24] and [25]. During the simulation, a strain at the for the C-S material model are presented in Fig. 9.
left-hand side of the sheet and the gross geometric The values of the cost function from Eq. (5b) for
226 Škrlec, A. – Klemenc, J.
 Strojniški vestnik - Journal of Mechanical Engineering 62(2016)4, 220-230

the J-C material model are presented in Fig. 10. In

both diagrams the logarithms of the cost functions
from Eqs. (5a) and (5b) are presented. The best three
combinations of the C-S parameters Et, P and C and
the J-C parameters B, n and c are listed in Table 4.
We can see from Fig. 9 and Table 4 that the best
combinations of the C-S parameters Et, P and C are
in the middle at the top of the original domain. The
best combinations of the J-C parameters B, n and c are
on the right of the original domains, see Fig. 10 and
Table 4. The parameter combinations with the worst
cost-function values failed to reproduce, in particular
the position of the maximum indentation depth during
simulations.
If we look at the results for both material Fig. 10. Cost-function values for the original domains of the
models, we can conclude that the optimal values of Johnson-Cook parameters
the individual parameters can be up to two orders of
magnitude distant from each other. This means that
the ranges for the original domains of both the C-S
and J-C parameter triples were too wide for a reliable
estimation of these parameters. For this reason we
decided to narrow the ranges of the C-S and J-C
parameter domains around their most promising
values from Table 4. This was followed by a new
simulation plan that was again composed with the
help of the L81(910) Taguchi array, but with the Et, P,
C and B, n, c parameter levels taken for the narrowed
domains in the same manner as was the case for the
original domain. The narrowed domains were as
follows: (i) for the C-S material model: C = 10 ms–1 to
46.4159 ms–1, P = 3.1623 to 10 and Et = 0.5623 GPa
to 1.7783 GPa; (ii) for the J-C material model: B =
0.1778 GPa to 3.1623 GPa, n = 0.1778 to 1 and c =
0.005623 to 0.1778.
Fig. 11. Cost-function values for the narrowed domains of the
Cowper-Symonds parameters

Fig. 9. Cost-function values for the original domains of the Fig. 12. Cost-function values for the narrowed domains of the
Cowper-Symonds parameters Johnson-Cook parameters

Estimating the Strain-Rate-Dependent Parameters of the Cowper-Symonds and Johnson-Cook Material Models using Taguchi Arrays 227
 Strojniški vestnik - Journal of Mechanical Engineering 62(2016)4, 220-230

We actually performed a “nested” Taguchi efficient for estimating the parameters of material
simulation plan in the second phase of the simulations. models that govern the material’s behaviour at high
The resulting distributions of cost-function strain rates. The added value of our approach is
values for the narrowed domains are presented in meaningful, especially in the cases when the number
Fig. 11 for the C-S parameters and in Fig. 12 for the of parameters that need to be identified is relatively
J-C parameters. The best three combinations of the high, with a wide range of potential parameter values.
C-S parameters Et, P and C and the J-C parameters With the Taguchi orthogonal array, a reasonable
B, n and c for the narrowed domains are much closer estimate of the material parameters can be found with
together when compared to the original domains for a relatively small computing effort and a short time.
those parameters. The average values from the three For example, if we applied a methodology that is based
best solutions are listed in Table 5. We can conclude on genetic algorithms and was originally developed
from Table 5 that our estimations of the parameters Et, for estimating the foam-material-model parameters
P, C and B, n, c are comparable to the values reported
[18], the processing times would be approximately
for mild steels in the literature ([9] to [11]), despite
100-times longer. This would be appreciated if the
different experimental arrangements and the fact that
numbers of parameters to be identified and the wide
the strain rates during the experiments were 10 or
ranges were to be increased.
more times higher in our case. We can conclude that
we obtained reasonable estimates of the parameters Et,
P, C and B, n, c for our case. 4 CONCLUSIONS
Plastic flow curves σ − ε effp for the two material
models are presented in Fig. 13. They were calculated The article presents a general approach to estimating
for different strain rates with the averaged parameters the parameters that govern a material’s behaviour
Et, P, C and B, n, c from Table 5. We can see from this at high strain rates. In our approach the Taguchi
figure that the estimates of the parameters for both experimental design was combined with the FE code
material models were consistent, because the flow LS-DYNA to estimate the material parameters using
curves span a similar domain of the σ − ε effp space for the results of the impact test between the ball and thin
the two material models. sheet metal. The presented approach was applied to
From the results we can conclude that the the realistic case of a material-parameter estimation
described methodology, which combines the nested for two different material models, i.e., the C-S and the
design with the FE simulations, can be very time J-C material models.

Table 4. The best three combinations for the original domains of the C-S and J-C material parameters
Parameter C [ms-1] Parameter P [/] Parameter Et [GPa] Cost-function value [/]
Cowper-Symonds Combination 1 21.5443 5.6234 1.0000 1.524
material model Combination 2 46.4159 10.000 1.0000 1.588
Combination 3 2.1544 3.1623 1.0000 1.826
Parameter B [GPa] Parameter n [/] Parameter c [/] Cost-function value [/]
Johnson-Cook Combination 1 0.1778 0.1778 1.0000 2.020
material model Combination 2 0.5623 0.4217 0.0056 2.259
Combination 3 1.7783 1.0000 0.0056 2.129

Table 5. The average of the best three combinations for the narrowed domains of the C-S and J-C material parameters
Parameter C [ms-1] Parameter P [/] Parameter Et [GPa] Cost-function value [/]
Our average 41.0133 6.2000 0.9550 1.522
Cowper-Symonds
Belingardi et al. [12] 3.006–4.987 1.329–1.619 - -
material model
Marais et al. [10] 2.000 5.000 - -
Markiewicz et al. [11] 1.150 7.750 - -
Parameter B [GPa] Parameter n [/] Parameter c [/] Cost-function value [/]
Johnson-Cook Our average 1.9250 0.8183 0.0972 2.138
material model Singh et al. [9] 0.779–2.692 0.743–0.928 0.0144–0.021 -
Marais et al. [10] 0.292 0.310 0.025 -

228 Škrlec, A. – Klemenc, J.

 Strojniški vestnik - Journal of Mechanical Engineering 62(2016)4, 220-230

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