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International Journal of Quantitative Research and
Modeling

e-ISSN 2721-477X
p-ISSN 2722-5046

Vol. 1, No. 2, pp. 59-66, 2020

A New Way to Identify Thin Sheet Behaviour: MicroInDef
Pierrick Malécot, Gemala Hapsari, Sébastien Thibaud, Fabrice
Richard
Univ. Bourgogne Franche-Comté, FEMTO-ST Institute, CNRS/UFC/ENSMM/UTBM, France
E-mail: pierrick.malecot@ens2m.fr, gemala.hapsari@femto-st.fr, sebastien.thibaud@ens2m.fr,

fabrice.richard@univ-fcomte.fr
.

Abstract
Incremental forming is a rapid prototyping process which uses a forming tool to form a sheet metal according to
a predetermined trajectory. In this work, a micro incremental deformation test (Micro InDef test) derived from
the principle of single point incremental sheet forming is developed and proposed. A complex mechanical
loading is applied and has a strong potential for the identification of inelastic behavior using inverse method. In
the first part, this work addresses the parameters identification and validation procedures of ductile damage
behavior of ultra-thin sheet metal under very large strain during this instrumented Micro InDef test. An inverse
finite element method based on the comparison between numerical and experimental axial forming forces of the
micro incremental deformation test is employed to extract a coupled ductile damaged plastic model. In the second
part, the objective is to prove the reliability of ductile damage parameters identification using forming force. The
richness of data contained in forming force is quantified and compared to the one from tensile test. Firstly, the
verification of the estimated parameter’s reliability is done via a simple analysis based on the forming force
sensitivity to material parameters and secondly by calculating elastoplastic and elastoplastic with ductile damage.
Keywords: Incremental forming, Micro InDef test, complex mechanical, finite element method.

1. Introduction
The mastery of numerical modeling is a preliminary step in processes optimizations and in
understanding complex phenomena. However, the quality of the numerical simulations depends on the
accuracy of the input data. Regarding thin sheet metal, the identification of constitutive law is often
performed by using classical characterization tests, i.e., tensile, bending, and shear tests. The strain level
reached by these tests is limited and does not represent the complex and large deformations that occur
during metal forming operations. The developed Micro Incremental Deformation (Micro InDef ) test
consists of locally deformed a clamped blank using a hemispherical tool. The advantages of this test are, the
magnitude of deformation achieved is considerably large and it can be performed on a CNC milling
machine, where the deformation path can be varied easily. Several studies have shown that the force
predictions made from finite element (FE) simulations of incremental sheet deformation are sensitive to
material parameters, for example Henrard et al. [1] and Duflou et al. [2] found that axial forming
force is the dominant force and is close to the total force. For these reasons we decided to use the axial
force as data for inverse problem approach using the finite element model
updating (FEMU)

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Parameters
E
ν
σY
Q
b
pD
S
s0
DC

Table 1. Material parameters
Definition
Young’s Modulus
Poisson’s ratio
Initial yield stress
Saturation value (Voce hardening law)
Hardening exponent (Voce hardening law)
Accumulated plastic strain threshold
Damage strength (Lemaitre denominator parameter)
Damaged material constant (Lemaitre exponent parameter)
Critical damage

Mechanisms
Elasticity
Elasticity
Plasticity
Plasticity
Plasticity
Damage
Damage
Damage
Fracture

method. The objective is to minimize the gap between a result response of a finite element
simulation and the measured quantity. Several authors have applied this method using global
measurements response, for example Gelin and Ghouati for an elastic plastic constitutive law
parameters using a deep-drawing test [3]. In this paper, the identification of a ductile damage
model is carried out (hereto a Lemaitre type behavior). A calibration procedure is proposed
to estimate the associated material parameters using the FEMU method. Then some tests
are presented to validate the inelastic behavior. Finally, a practical identifiability analysis is
performed to quantify the information richness of forming forces measurement.
2. Material and methods
The selected material for this study is a single-phase copper foil with an initial thickness of 210
µm. The material is annealed at 400◦ C for 30 min to eliminate the effects of rolling texture
and to refine the structure by making it homogeneous. The average grain size after annealing is
equal to 30 µm. The elastic parameters are obtained from ultrasonic characterization . Uniaxial
tensile tests were conducted on flat specimens in three directions: 0◦ , 45◦ and 90◦ with respect to
the rolling direction. The specimen was elongated up to fracture, and the true stress-strain curve
was obtained for the three orientations (0◦ , 45◦ and 90◦ ). The calculated normal anisotropy RN
and the planar anisotropy are RN =0.955 and ∆R=-0.12, respectively. These results demonstrate
that the material can be considered isotropic. Due to experimental results, the sensitivity to
the strain rate is not considered.
2.1. Material and considered behavior law
The continuum damage mechanics concepts developed by Lemaitre and Desmorat [4] in the
context of thermodynamics with internal variables are considered. 9 parameters should be
identified, as listed in Table 1. These parameters are divided into four categories: characterizing
the elasticity, plasticity, damage and fracture mechanisms.
2.2. Calibration methods of the coupled damage-plasticity model
The idea is to compare an experimental Micro InDef test and a numerical version (FEM) of
the same test. More precisely we compare the experimental and numerical forming forces on Z
axis. The material parameters are estimated in order to minimize the difference between these
two forces. The procedure is divided in 2 steps. First step (initialization) is the identification
of elastic and plastic parameters by comparing results between tensile testing and modeling.
The result of first step is then used as input data for the second step (estimation), which is

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61

identification of plastic and damage parameters by comparing forming forces between Micro
InDef testing and modeling.
2.3. Tensile test (Step 1)
The first step is dedicated to determinate the initial hardening parameters set (σy0 ,Q0 ,b0 )
by using tensile test. This step is necessary to start the identification (step 2) from a
physically solution and to reduce the computation time. The first step comprise of experimental
and numerical procedure. Experimental quasi-static tensile tests are performed. A finite
element parametric model is then used to simulate tensile test. Initial hardening parameters
determination for the tensile test is realized using the FEMU method. Then, the numerical
reaction force is compared to the experimental measurement and the hardening parameters are
adjusted iteratively using an optimization algorithm. In order to avoid the necking phenomenon,
only the experimental data until 23% of deformation are considered. A constrained optimization
algorithm based on the Levenberg-Marquardt method is used to solve the problem in the MIC2M
software. More details are given in [5].
2.4. Micro Incremental Deformation (Micro InDef ) test (Step 2)
The Micro InDef testing device, illustrated in Figure 1.a., is composed of a fixed die support,
a modular die, a fixed blank holder clamped to the die using screws and a forming tool with a
radius of 1 mm (hemispherical end tool). The lubrication of the sheet/tool interface (water/oil
mixture) is used to improve the sheet formability and to minimize frictions effects. The tool
moves with a constant feed rate of 500 mm/min and rotates with a constant speed rate of 1000
rpm to ensure the spindle integrity and reduce the effects of friction. A 3-axis micro-milling
CNC Machine (KERN) is used and the forming forces Fexp are acquired by using a 4-axis
dynamometer (Kistler 9272), as represented in Figure 1.b.

Figure 1. Testing device: (a) Micro InDef testing device, (b) forces acquisition principle(Fexp )
As for the tensile ones, each test is repeated three times to ensure that the test is reliable.
Pyramidal shape, with a draft angle α, is used to perform the Micro InDef test. The geometrical
definition of this shape is given in Figure2.
A fully parametric toolbox, programmed in MATLAB language, has been developed to
prepare the input files necessary for Micro InDef test’s simulation (mesh, boundary, load and

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Figure 2. Definition of the pyramidal part shape.L1 = L2 = 30mm, l1 = l2 = 6mm, H = 4mm,
R = 1mm, α = 2◦
initial conditions, material behavior). This numerical toolbox was completely detailed and
validated in the study proposed by [6]. Considering the numerical approach, the blank is meshed
with 8-nodes fully integrated solid elements and tools with 4-nodes rigid shell elements. Three
elements are used in the blank thickness (6 integration points through the thickness). 120
elements are imposed upon the length and width to discretize the blank with a total of 43200
solid elements. The tool and the blank are modeled with the same geometrical parameters as
in the experimental test. During simulation tests, the sheet is clamped along its edges. The
LS-DYNA software with an explicit integration method is used to simulate the Micro InDef test.
The coupled damage-plasticity model used for this simulation represents the main mechanisms of
inelastic behavior, including plastic deformation, the change of elastic response and the localized
failure.The friction law chosen to simulate the tribological behavior at the interfaces between
tools and the blank is a Coulomb’s friction law, with a friction coefficient equal to 0.2. This
choice derives from studies proposed by [6] on the influence of friction on forming forces level.
To summarize : this step consists of simulating a Micro InDef test, whose results (forming force)
are sensitive to material parameters. These material parameters are the ones to adjust. The
test is first simulated with a set of initial parameters identified from tensile test, considered to
be physically acceptable. Then, numerical results are compared to experimental measurements
and the material parameters are adjusted iteratively using the same optimization algorithm as
on tensile test. The initial values of the accumulated plastic strain threshold p( D0 ), the damage
strength s0 and the damaged material constant t0 are introduced from the literature. Here, the
cost function ω2 (θ 2 ) is defined as the gap between the axial forming forces obtained through the
experiments FZE (t) and those obtained through the simulation, FZN (θ 2 ,t) of Micro InDef test
with the helical strategy.

ω2 (θθ 2 ) =

N2

2
1 X
F ZE (ti ) − F ZN (θθ 2 ,t i )
N2

(1)

i=1

Where N2 =1000 is the number of experimental points, which is equally distributed over the
time interval[t1 ,tN2 ], and θ2 is the vector of material parameters. Six parameters associated with
plasticity and damage mechanisms θ̂2 =T (σ̂y ,Q̂,b̂,p̂D ,Ŝ,ŝ0 ) which minimize this cost function, are
thus estimated. The critical damage parameter Dc is estimated after the minimization procedure
by detecting the moment of the experimental fracture. For each iteration of the calibration
procedure, if n represents the number of parameters to identify, it should be performed at least
n+1 simulations. This is necessary to fulfill the optimization algorithm requirements. Due to the
significant size of the problem and the stability condition associated with the explicit algorithm,
simulation time holds an important role, for this study one simulation requires approximately 8
hours. The massively parallel processing (MPP) version of LS-DYNA is used with 16 processors
to decrease the computational time.

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63

2.5. Validation tests
A validation procedure is carried out to verify the quality of the calibrated model. Firstly,
the experimental part geometry is compared with the one obtained with the finite element
simulation. The experimental part is digitalized using a non-contact high resolutin 3D laser
scanning system. The experimental data are then compared with the mesh using inspection
software (Geomagic Qualify). Secondly, validation tests are done. The coupled damage-plasticity
model identified with the helical strategy and pyramidal geometry with the procedure defined
previously is for example used to simulate:
• pyramidal-shaped part using a Z-level strategy.
• conical shaped part using both forming strategies (helical and Z-level strategies).
• FLD are simulate and compared to experimental ones
3. Quantification of information richness from forming force
An identifiability analysis is conducted to ensure the parametric identification problem is wellposed, FEMU method robustness and physical significance of the parameters. This analysis uses
a scalar criterion, based on the forming force sensitivity vectors’ collinearity and norm. It allows
to quantify information richness and all parameters’ subset identifiability. This type of analysis
has already been applied in other disciplines that develop over-parameterized models, e.g., [7]
and [8] for the environmental simulation models and econometrics, respectively. Recently, [9]
employed this technique to ensure the identifiability of viscoelastic behavior from instrumented
spherical indentation test.
A criterion based on axial forming force sensitivity, is used to measure the parameter
identifiability of the coupled damage-plasticity constitutive law. The purpose is to determine
the sensitivity of axial forming force numerical simulation during Micro InDef test for various
input material parameters. The parameter sensitivities are computed using the backward finite
difference method. The components of the dimensionless sensitivity matrix S are mathematically
defined by:

Sij =

θ2j
∂FZN
FZN ,max ∂θ2j

(2)

For computing this matrix, the optimization software MIC2M is used and a sensitivity ranking
is performed by means of the following relation:

N

δj =

1 X
|Sij |
N

(3)

i=1

where N is the number of measurement points. The sensitivity ranking measures the mean
sensitivity of the simulated z-axis forming force to a variation of the parameter value θ2j . A
high δj means that the value of the parameter θ2j has an important influence on the simulation
result, while a value of zero indicates that the simulation result does not depend on the parameter
θ2j .
A local identifiability index IK of parameter subset K can be written as follows [10]:


IK = log10

λmax
λmin


(4)

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λmax and λmin are the largest and the smallest eigenvalue of the dimensionless pseudo-hessian
matrix H, respectively. The H matrix is estimated from a dimensionless sensitivity matrix
(defined for a subset K of parameters) by the following relation:

H = ST S

(5)

I K smaller than 2 are considered as having high identifiability, while a value above 3 indicates
a low identifiability. The I K index is computed for all the subsets of the parameter space and
reveals all the identifiable sets of parameters (I K < 3) by quantifying the richness of the used
data.
4. Results and discussion
4.1. From Step 2 (Micro InDef Test)
By using the parameters estimation from Micro InDef test, the parameters are obtained in eight
iterations. The set of calibration parameters is summarized in Table 2.
Table 2. Material parameters calibrated by using Micro InDef test with helical strategy and
pyramidal geometry.
Hardening
Damage
Fracture
Q̂
Ŝ
D̂c
b̂
σˆy
p̂D
ŝ0
67.97 MPa 189.60 MPa 16.00 0.35 1.31 MPa 1.01
0.68

The comparison between the numerical axial forming FZ and the experimental one is
presented in Figure 3

Figure 4. Numerical and experimental
Figure 3. Evolutions of numerical and comparisons: (a) section, (b) thickness
experimental axial forming forces.
evolution.

4.2. validation tests
To validate the ductile damage model, a comparison between the final numerical geometry and
the experimental one is carried out. The pyramidal shape obtained via simulation is close to the
experimental part. The differences between the edges of workpiece (±0.15 mm) and the base of
pyramid (±0.06 mm). The value of die radius and its position strongly influence the final shape.
The actual die radius is not strictly identical to that used for the simulations. Moreover, this
area is highly dependent on the springback effect, which is a difficult parameter to control. The
comparison between the numerical model section and the experimental one is given in Figure 4a,

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65

and the thickness evolution is plotted in Figure 4b. A good correlation between experiments
and simulations can be noticed. The minimum thickness value is approximately 0.07 mm, which
corresponds to a thinning of 66%.
The parameters presented in Table 2, are used to simulate different tests. For the forming
of a conical geometry with helical strategy, a good agreement between the numerical prediction
and the experimental measurements in terms of forming forces (level and tendency) geometric
accuracy and fracture prediction is noted (an example is given Figure 5).

Figure 5. Evolutions of numerical and experimental axial forming forces using the helical
strategy and conical geometry.
Other validation tests are presented in Hapsari [5].
4.3. Forming force information richness
Figure 6 compares tensile force and axial forming force sensitivity δj to the plastic damage
parameters. The sensitivity is defined by Equation 3. It can be seen that tensile force is sensitive
to plastic parameter variations, but is very insensitive to damage parameters. Concerning the
axial forming force, it is important to note that all sensitivities are approximately of the same
order of magnitude. This is a necessary condition (but not sufficient) for good conditioning of
the parametric identification problem.

Figure 6. Average sensitivities of the
Identifiability index using
tensile force and the forming force to the Figure 7.
tensile
test
and
Micro InDef test for
plastic damage parameters using tensile
different
parameters
subsets K.
and Micro InDef test.
Figure 7 shows the measurement of identifiability index (Equation 4 for three sets of
parameters K : plastic (σ̂y ; Q̂; b̂), damage (p̂D ; Ŝ;Ŝ0 ) and plastic with damage (σ̂y ; Q̂; b̂ ;p̂D ; Ŝ;Ŝ0
) for tensile and Micro InDef. In the case of tensile test, the K values show that plastic sets are

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identifiable (K < 2) and the damage parameters are weakly identifiable (K > 3). These results
show the poorness of experimental data obtained by tensile test when identifying the damage
parameters. On the contrary, for Micro InDef test with helical strategy, all the parameter sets
lead to low values (K < 2). It can be then considered as a promising technique to accurately
identify material in very large deformations.
5. Conclusions
This work is dedicated to the complete definition of the identification and identifiability of the
ductile damage behavior of ultra-thin sheet metal under notably large strain via the Micro InDef
test.
The following conclusions can be drawn from the present work:
• The initial plastic material parameters obtained from the tensile tests are close to the final
calibration. It is impossible to identify the damage law from only tensile test results because
the failure is due to both mechanisms: damage growth and necking.
• The results of the validation tests, using the parameters estimated via FEMU method,
showed good agreement compared with the experimental measurements.
• The identifiability analysis demonstrates the advantage of using instrumented deformation
process for determining the meaningful mechanical properties of thin sheets under very
large strain (in regards to the material in this study, approximately 240
Future studies will focus on the application of the proposed test as a characterization method
with more complex models. typically, the effect of anisotropy, kinematic hardening or multiple
hardening and more complex damage laws will be considered.
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