B. SENER, H. KURTARAN: MODELING THE DEEP DRAWING OF AN AISI 304 STAINLESS-STEEL ...
961–965
MODELING THE DEEP DRAWING OF AN AISI 304
STAINLESS-STEEL RECTANGULAR CUP USING THE
FINITE-ELEMENT METHOD AND AN EXPERIMENTAL
VALIDATION
MODELIRANJE GLOBOKEGA VLEKA PRAVOKOTNE ^A[E IZ
AISI 304 NERJAVNEGA JEKLA Z METODO KON^NIH
ELEMENTOV IN Z EKSPERIMENTALNIM PREVERJANJEM
Bora Sener1, Hasan Kurtaran2
1Yildiz Technical University, Department of Mechanical Engineering, Istanbul, Turkey
2Gebze Technical University, Department of Mechanical Engineering, Kocaeli, Turkey
borasener84@gmail.com, borasen@yildiz.edu.tr
Prejem rokopisa – received: 2015-09-06; sprejem za objavo – accepted for publication: 2015-11-16
doi:10.17222/mit.2015.278
In this paper the deep drawing of a rectangular cup from AISI 304 stainless steel sheet was investigated numerically and experi-
mentally. The finite-element method was used for computer modeling of the deep-drawing process. The thickness distribution
predicted from the finite-element analysis was compared with experimental measurements. It was observed that the numerical
results agree well with the experimental values. The minimum thickness was observed at the punch radius in both the simulation
and experiment.
Keywords: deep drawing, rectangular cup, finite element, stainless steel
V ~lanku je bil numeri~no in eksperimentalno preiskovan globoki vlek pravokotne ~a{e iz plo~evine iz nerjavnega jekla AISI
304. Za ra~unalni{ko modeliranje procesa globokega vleka je bila uporabljena metoda kon~nih elementov. Iz analize kon~nih
elementov napovedana razporeditev debeline je bila primerjana z eksperimentalnimi meritvami. Preiskava je pokazala, da se
numeri~ni rezultati dobro ujemajo z eksperimentalnimi vrednostmi. Najmanj{a debelina je bila opa`ena na radiusu pesti~a, tako
pri simulaciji kot tudi pri eksperimentu.
Klju~ne besede: globoki vlek, pravokotna ~a{a, kon~ni element, nerjavno jeklo
1 INTRODUCTION
The rectangular/square-cup deep-drawing process has
specific forming characteristic. Non-uniform material
flow and quite complicated deformation mechanism are
seen in this process. Therefore, the deep drawing of
square and rectangular cups is more difficult than that of
some other shapes, such as circular cups. Many resear-
chers investigated the rectangular/square-cup deep-draw-
ing process experimentally and numerically. A. G. Ma-
malis et al.1 investigated the effect of material and
forming characteristics on the simulation of the deep
drawing of square cups by using the explicit non-linear
finite-element code DYNA-3D. They considered the
effect of material density, punch velocity and friction
coefficient. L. F. Menezes and C. Teodosiu2 studied the
square-cup deep-drawing process numerically and expe-
rimentally. They modeled the process by using solid
elements and compared the numerical results with the
experiment. E. Bayraktar and S. Altintas3 investigated
the square-cup deep-drawing process and 2D-draw
bending process of Hadfield steel experimentally. They
evaluated the draw-in values of the flange, the principal
strains in the square-cup deep drawing and compared the
experimental results with that those of mild steel. Y. Ma-
rumo and H. Saiki4 studied differential lubrication
methods in the square-cup deep-drawing process in order
to prevent any deformation concentration on the corners.
Y. Harada and M. Ueyama5 investigated the drawability
of pure titanium sheets in the square-cup deep-drawing
process. Titanium sheets were coated by heat oxide and
formed into a square with a punch. L. M. A. Hezam et
al.6 developed a new technique for the deep drawing of
square cups made from brass and pure aluminum. They
improved the material flow by using a conical die with a
square aperture at its end without a blank holder. M.
Gavas and M. Izciler7 designed a blank holder with a
spiral spring to reduce the friction area between the
blank and the blank holder during the deep drawing of
square cups. A higher drawing height, a homogenous
thickness distribution and minimum earing cups were
obtained in their study. M. A. Hassan et al.8 have deve-
loped a new divided blank holder with a tapered base and
eight tapered segments to increase the deep drawability
of the square cups. They improved the drawability of
thin sheets and foils and increased the limiting drawing
ratio with this technique over the conventional techni-
ques. L. P. Lei et al.9 studied the square-cup deep-draw-
ing process for 304 stainless-steel sheet numerically and
experimentally. They evaluated the effect of the blank
Materiali in tehnologije / Materials and technology 50 (2016) 6, 961–965 961
UDK 621.77:004.946:669.14.018.8:620.1 ISSN 1580-2949
Original scientific article/Izvirni znanstveni ~lanek MTAEC9, 50(6)961(2016)
shape on the material flow. J. H. Lee and B. S. Chun10
investigated the effect of temperature, blank shape and
holding pressure on the deep drawability of a square cup
from 304 stainless-steel sheet experimentally and nume-
rically. F. K. Chen and S. Y. Lin11 examined the effects of
process parameters such as punch radius, die radius, die
corner radius, die gap and the length-to-width ratio by
both the finite-element method and the experimental
approach. The authors formulated a formability index
that serves as a design rule for the rectangular cup draw-
ing from 304 stainless-steel sheet. Although many signi-
ficant studies are carried out about the rectangular/
square-cup deep-drawing process, they are generally
limited to the forming of Al and Al alloys. Very little of
the literature is devoted to the rectangular-cup deep
drawing of 304 stainless-steel sheets. The available
literature on the rectangular-cup deep drawing of 304
stainless steel sheets is limited to warm forming.
Therefore, the cold forming of a rectangular cup from
304 stainless steel sheet is investigated numerically and
experimentally in this study.
2 EXPERIMENTAL PROCEDURE
A die with a rectangular aperture, a conical punch
that has flat surface with a rectangular shape and a
circular blank holder that has a rectangular cavity are
used in this study. The dimensions of the rectangular-cup
tooling are given in Table 1. Deep-drawing experiments
were carried out on a 160-ton-capacity double-action
hydraulic press. The punch is mounted to the lower shoe.
The blank holder slides around the punch in the lower
shoe. The upper shoe consists of the die. In the press, the
lower shoe is mounted on the press bed and the upper
shoe is attached to the press ram. The blank holder is
supported by the cushion pins that apply the blank hold-
ing force to the blank holder during the forming process.
During the deep-drawing process, the blank holder is
raised to the top-most level. The blank is positioned on
the blank holder. The clamped blank with the die and
blank holder moves further down and forms the blank
against the stationary punch under the action of the blank
holder force through the cushion pins. The experimental
set-up is shown in Figure 1.
Table 1: Tool dimensions
Tabela 1: Dimenzije orodja
Die
Radius, mm 35
Edge length of rectangular cavity, mm 129 × 143
Corner radius of the rectangular cavity, mm 47
Depth of the rectangular portion, mm 27
Punch
Radius, mm 20
Rectangular side length, mm 120 × 134
Cone angle, deg 2
Blank
holder
Diameter, mm 351
Edge length of rectangular cavity, mm 127.5 × 142
Corner radius of the rectangular cavity, mm 45.5
An austenitic grade AISI 304 stainless steel was used
in this study. The thickness of the material was 0.8 mm.
The mechanical properties of the material are explained
in Section 3. The initial blank of diameter 335 mm was
drawn to a rectangular cup of height 80 mm. A 340-kN
blank holder force was applied in the experiments. The
operating speed was 20 mm/s. A rectangular cup from
304 stainless-steel sheet was successfully drawn using
the this blank holder force as shown in Figure 2.
3 FINITE ELEMENT MODEL
In the present work, the explicit non-linear finite-ele-
ment (FE) code DYNAFORM 5.9.2 software is used for
B. SENER, H. KURTARAN: MODELING THE DEEP DRAWING OF AN AISI 304 STAINLESS-STEEL ...
962 Materiali in tehnologije / Materials and technology 50 (2016) 6, 961–965
Figure 2: Rectangular cup
Slika 2: Pravokotna ~a{a
Figure 1: Experimental set-up
Slika 1: Eksperimentalni sestav
simulating the rectangular-cup deep-drawing process.
The blank is meshed with 3333 quadrilateral elements
and 3434 nodes. A Belytschko-Tsay shell element with
five integration points across the thickness is used for the
shell mesh of the blank. Because of the symmetry, only a
quarter model is employed in the numerical simulation,
as shown in Figure 3. The punch, die and the blank
holder were modeled as rigid objects because of their
high stiffness, while the blank was modeled as a deform-
able body. A forming-one-way-surface-to-surface con-
tact algorithm is used in the analysis. The friction coeffi-
cients are assumed to be 0.11 for the contact between the
tools (die, punch and blank holder) and the blank. This
value was recommended by a previous investigation.12
The die speed employed is 1000 mm/s, which is extrem-
ely slow compared to the typical wave speeds in the ma-
terials to be formed (the wave speed in steel is approxi-
mately 5000 m/s). In general, inertia forces will not play
a dominant role for forming rates that are considerably
higher than the nominal 1000 mm/s rates in the physical
problem.13 The displacement of the die was taken as
80 mm, which is decided by the height of the cup. A
85-kN constant blank holder force is applied in the
model (quarter of the experimental value). AISI 304
stainless-steel sheet is used in the simulation work. It is
assumed that the material is isotropic and homogeneous.
The strain-hardening model used is isotropic hardening.
The mechanical properties of the material were
determined by tensile testing. Different constitutive
equations, such as Holloman, Swift, and Ludwick, were
evaluated in order to represent the plastic behavior of the
AISI 304 stainless steel. The nonlinear least-squares
method and a trust-region algorithm were used in
determining the material parameters. It was found that
the Ludwick equation was the best fit to the experimental
data for the AISI 304 stainless steel. This result agrees
with the literature14. The flow curve was extrapolated to
higher strains using this equation and was used in the
simulation. A comparison of these different hardening
models with experimental data is shown in Figure 4.
The mechanical properties of the materials are
reported in Table 2.
Table 2: Mechanical properties of the material
Tabela 2: Mehanske lastnosti materiala
Parameters Units Value
Young’s modulus (E) GPa 256.86
Poisson’s ratio (v) 0.28
Yield strength (y) MPa 308.94
Strain hardening (n) 0.3
Coefficient of strength (K) 1528
4 RESULTS AND DISCUSSION
The thickness distribution is one of the major quality
characteristics in the sheet-metal formed part. Therefore,
the thickness distribution of the AISI 304 stainless-steel
sheet in the deep-drawing process was investigated
theoretically and experimentally. The drawn component
was cut along the diagonal direction and the thickness of
the part along this direction was measured using a micro-
meter, as shown in Figure 5.
For the verification of the FEM results, the thickness
variations predicted by the numerical simulation were
compared with the experimental results. Figure 6 shows
the comparison of the FE predictions with the experi-
ment for the formed part along section YO (diagonal). It
could be observed from Figure 6 that the thickness
B. SENER, H. KURTARAN: MODELING THE DEEP DRAWING OF AN AISI 304 STAINLESS-STEEL ...
Materiali in tehnologije / Materials and technology 50 (2016) 6, 961–965 963
Figure 5: Location of the diagonal section in the formed part for
evaluation of the thickness distribution
Slika 5: Prikaz diagonalnega preseka preoblikovanega dela za oceno
razporeditve debeline
Figure 3: Finite-element model
Slika 3: Model kon~nega elementa
Figure 4: Comparison of the hardening models with experimental
data
Slika 4: Primerjava modelov utrjevanja z eksperimentalnimi podatki
distribution predicted by the FE simulation along section
YO agrees with the experiment.
A minimum thickness was observed at the punch
radius for both the simulation and the experiment. This is
due to the appearance of the biaxial stretching state in
this region. Hence, it is obvious that the potential failure
site in the deep drawing of the part is located in the
vicinity of the punch radius where the thinning is a maxi-
mum. This phenomenon is also observed by S. Gallée12
during the deep drawing of stainless-steel sheet.
Although the trend of thickness distribution predicted
by the simulation matches with the experiment, more
thinning was observed in the simulation (32 %) than in
the experiments (26 %). The difference in thinning could
be attributed to the insufficiency of the tensile test. The
flow stress available from the tensile test was limited to
small strains at low strain rates. The flow curve of the
material was obtained up to the 0.21 plastic strain value
in the tensile test. Beyond this value, the hardening curve
was extrapolated to high strains using the Ludwick equa-
tion. These values were probably overestimated. Thus,
more restraining force to draw the material from the
flange resulted in a localized thinning of 32 % in the
simulation.
5 CONCLUSIONS
Finite-element simulations and deep-drawing experi-
ments of the AISI 304 stainless-steel sheet were carried
out. The conclusions can be summarized as follow:
• Different constitutive equations were evaluated to
represent the plastic behavior of the AISI 304
stainless steel sheet. The Ludwick equation was the
best fit to the experimental data for this steel.
• Good agreement was obtained between the experi-
mental and finite-element results. Maximum thinning
was observed at the punch radius in both the simula-
tion and experiment. This is because of the appear-
ance of the biaxial stretch state at the punch radius.
• Predicted thinning values were larger than the experi-
mental data at the punch radius. More thinning
observed in the simulation could be due to the extra-
polation of the flow stress to higher strains in the
software and overestimated.
Acknowledgements
The authors would like to thank Mr. Kani Yilmaz for
his help with the experimental set-up.
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964 Materiali in tehnologije / Materials and technology 50 (2016) 6, 961–965
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