REGULAR ARTICLE
Numerical and experimental study of bursting prediction in
tube hydroforming of Al 7020-T6
Arvand Afshar, Ramin Hashemi
*
, Reza Madoliat, Davood Rahmatabadi, and Behzad Hadiyan
School of Mechanical Engineering, Iran University of Science and Technology, Tehran, Iran
Received: 24 November 2016 / Accepted: 6 March 2017
Abstract. In this study, forming limit diagram (FLD) of tubular material (Al 7020-T6) was determined
numerically and experimentally. A set of experimental bulge tests were carried out to determine FLD under
combined internal pressure and axial feeding. Also, a numerical approach which is based on the acceleration of
plastic strain (i.e., the second derivation) was applied to compute the hydroforming strain limit diagram. Based
on this method, the localized necking would be started when the acceleration of the max plastic strain got its
maximum value. Finally, the numerical FLD was verified by experimental test results on the aluminum tube
7020-T6 and a good agreement between the proposed method and the experimental works was observed.
Keywords: tube hydroforming / bursting / second derivative of strain / forming limit diagram / experiment
1 Introduction
Forming limit diagram (FLD) is a significant criterion for
evaluating formability of tubular materials, which is
commonly obtained from theoretical calculations, finite
element simulation and experiment. Hydroforming process
makes integrated tubular parts with high ratio of strength-
to-weight in one step. By applying oil pressure into the
tube, and applying axial force to the ends, a tubular blank
is formed into the internal shape of the die. In this process,
the original specimen is a simple tube (direct or bend tube).
Due to increasing requests for light-weight parts, hydro-
forming processes have been widely used to produce and
make parts in various fields, such as automobile, aircraft,
aerospace, and shipbuilding industries [1]. Hashemi et al.
[1] have also considered tube hydroforming process,
including the manufacturing of metal bellows.. Asnafi
and Skogsgårdh [2] proposed a mathematical model to
predict the forming pressure and the related feeding
distance required to hydroform a circular tube into a T-
shape product without wrinkling and bursting. The use of
aluminum alloys in the place of steel components in
automotive applications saw a significant increase during
the last few years. For this reason, hydroforming of
aluminum tubes is a very desirable manufacturing process
instead of sheet metal forming. In tube hydroforming, it is
required that the vacant tube should be formed into a die
cavity of the final shape without any kind of de ficiency such
as bursting, wrinkling or buckling. Since bursting is an
impression of localized necking which is a condition of local
instability under excessive tensile stresses, prediction of
necking is an important problem before designing the
details of processes [3]. FLDs are appointed to determine
the tubular materials formability. The laboratory test
results showed that the FLDs are influenced by several
parameters including the strain rate [4 ], strain hardening
and anisotropy coefficients [5], heat treatment [6], grain
size [7] and strain path changes [8]. After obtaining the
forming limit curves (FLCs) by Keeler and Backofen [9],
many researchers tried to develop some numerical and
analytical models to determine the sheet metal formability.
But, only a little attention has been paid to study the
behavior of tubular materials. For example, Kim et al. [10]
predicted the bursting failure in tube hydroforming
considering plastic anisotropy by using numerical calcu-
lations. Song et al. [11] used analytical approach to
bursting in tube hydroforming using diffuse plastic
instability. One year later, the team combined the two
previous methods: analytical and numerical methods for
prediction of forming limit in tube hydroforming [12].
Hwang et al. [13] predicted FLDs of tubular materials by
bulge tests in two ways. They have used the Hill’s law for
calculations and did bulge tests for an experiment and then
compared the two methods together. Chen et al. [14] used
thickness gradient criterion for seamed tube hydroforming
that resulting in FLD. They validated numerical solution
with experimental work. Seyedkashi et al. [15] analyzed
two-layered tube hydroforming with analytical and
experimental verification.
* e-mail: rhashemi@iust.ac.ir
Mechanics & Industry 18, 411 (2017)
© AFM, EDP Sciences 2017
DOI: 10.1051/meca/2017019
Mechanics
&
I
ndustry
Available online at:
www.mechanics-industry.org
In this paper, FLCs of tubular materials (Al 7020-T6)
with respect to axial feeding and hydraulic pressure were
determined numerically and experimentally for the first
time. The computed FLD was verified by a series of
experimental bulge tests. A numerical approach was
applied to FLC prediction. This numerical method is
based on the acceleration of plastic strain (i.e., the second
derivation) which was applied to determine the onset of
necking for tube materials. Based on this method, the
localized necking would be started when the acceleration of
the max plastic strain gets its maximum value.
2 Experimental work
2.1 Tube bulging test
The dimensions and configurations of an initial tube and its
final bulged part are shown in Figure 1 . The outer diameter
of the pipe was 40 mm and the initial thickness of the
tubular blank was 1.5 mm. Aluminum pipes were seamless
and produced by extrusion process.
The mechanical and material properties of the tube
were determined by standard test using specimen, which
were prepared according to ASTM-E8 specification at a
constant crosshead speed of 2 mm min
1
. The mechanical
and material properties are presented in Table 1.
To evaluate the hydroforming limit strain diagram, a
series of bulge tests were carried out on aluminum tube
7020-T6. For doing the tests, an experimental setup with
the ability to control internal pressure and axial feeding
was provided. This setup had two hydraulic jacks and a
hydraulic pump and it is shown in Figure 2. All hydraulic
instruments used in the experimental procedure, including
Fig. 1. Dimensions and configurations of an initial tube and its final bulged part (mm).
Table 1. The material and mechanical properties of aluminum tube 7020-T6.
Material Specific
gravity (kg m
3
)
Young module,
E (GPa)
Yield strength,
YS (MPa)
Strength coefficient,
K (MPa)
Strain hardening
index, n
Al7020-T6 2780 71 305 370 0.17
Oil inlet
Axial feeding Dies
Fig. 2. Free bulge setup.
Fig. 3. The input loading paths with a combination of internal
pressure and axial feeding.
Fig. 4. Circles engraved on the tube.
2 A. Afshar et al.: Mechanics & Industry 18, 411 (2017)
pumps and valves were fabricated in the Enerpac company
(the supplier’s name was Enerpac). The measurement
accuracy of hydraulic pump was up to 1 bar. The two ends
of the tube were free to be able to move in the axial
direction for providing axial feeding. Internal pressure
measured by barometer and axial feeding measured by
linear variable differential transformer with the measure-
ment accuracy of 0.01 mm.
To obtain the FLCs, different loading paths with a
combination of internal pressure and axial feeding s hould
be applied t o the tube. For this purpose, the linear loading
curves from internal pressure and axial feeding were used.
The six applied load paths are shown in Figure 3. Loads
were applied in two steps, initially the internal pr essure
was increased and then the axial feeding was applied, till a
burst occurred in the tube. The internal pressure and the
axial feeding displacements (e.g., the input loading paths)
show n in Figure 3 which were controlled by the PC-based
controller of the experimental setup for a series of the
bulging tests. For measuring strains in the experimental
work, a regular grid layout of the circle with a diameter of
2.5 mm on the samples was etched. To carve these circles,
electrochemical etching device was used.
The circles engraved on the tube were shown in
Figure 4. After examination of the bulge tests, the circles
transformed to ellipses after deformation. The major and
minor diameters of the ellipses were measured using a
profile projector machine.
As a result of excessive pressurizing during the bulge
process, bursting occurred in the middle of the tube wall as
illustrated in Figure 5.
To determine hydroforming strain limit diagram
experimentally, at first, the tubes were carved and then
placed under loadings. Loadings stopped when the tube
burst occurred. After the bursting, the major and minor
diameters of the ellipses near the crack were measured and
then the limit strains were calculated. The major and minor
engineering strains can be obtained from the following
equations. Measuring diameters were performed by using
the profile projector machine.
e
1
¼
a d
d
ð1Þ
e
2
¼
b d
d
ð2Þ
In these equations, “a” is the large diameter of ellipse and “b”
is the small diameter of it. “d” is the diameter carved in
advance.
3 Finite element modeling
The ABAQUS/Explicit FE software was used to model the
hydroforming process in order to investigate the FLDs of
aluminum tubes. All the analyses were realized using an
explicit finite element approach. The die map used in the
simulation can be seen in Figure 1.
(a) Without axial feeding
(b) Axial
feeding= 2mm
(d) Axial
feeding= 6mm
(c) Axial
feeding= 4mm
Fig. 5. Experimental bursting failure obtained from the bulge tests under the different loading paths: (a) without axial feeding,
(b) axial feeding = 2 mm, (c) axial feeding = 4 mm, and (d) axial feeding = 6 mm.
Fig. 6. Finite element simulation model.
A. Afshar et al.: Mechanics & Industry 18, 411 (2017) 3
This process was simulated with the solving dynamic/
explicit. Material properties were extracted with use of
uniaxial tensile tests and were entered in the relevant
module. Penalty method was used to establish contact
between the tube and mold. Anisotropy coefficients for that
material, by the simple tensile test were measured in
different directions. To apply anisotropy into the simula-
tion, the Hill ’ s 48-yield criterion [16] was used. Hill’s 48-
yield criterion and its coefficients based on the measured
anisotropy in the directions of 0, 45 and 90 in equations (3)–
(7) is given.
see equation (3) below
H ¼
r
0
1 þ r
0
ð4Þ
F ¼
H
r
90
ð5Þ
G ¼
H
r
0
ð6Þ
N ¼
ðr
90
þ r
0
Þð2r
45
þ 1Þ
2r
90
ð1 þ r
0
Þ
ð7Þ
The coefficients of Hill’s 48-yield criterion for a three-
dimensional stress mode and its relation with the main
factors yield criterion are given below.
F ¼
1
2
1
R
22
2
þ
1
R
33
2
1
R
11
2
ð8Þ
Fig. 7. Strain distributions for 4 mm axial feeding: (a) max strain and (b) min strain.
Fig. 8. The relationship between the two criteria large strain and
small strain.
Fig. 9. The highest and lowest strain versus time for plane strain
mode.
Fig. 10. The second derivative of max strain for 4 mm axial
feeding.
f
ðÞ
¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Fð
22
33
Þ
2
þ Gð
33
11
Þ
2
þ H ð
11
22
Þ
2
þ 2L
23
2
þ 2M
31
2
þ 2N
12
2
q
ð3Þ
4 A. Afshar et al.: Mechanics & Industry 18, 411 (2017)
G ¼
1
2
1
R
33
2
þ
1
R
11
2
1
R
22
2
ð9Þ
H ¼
1
2
1
R
11
2
þ
1
R
22
2
1
R
33
2
ð10Þ
L ¼
3
2R
23
2
ð11Þ
M ¼
3
2R
13
2
ð12Þ
N ¼
3
2R
12
2
ð13Þ
In this paper, for convenience, a Cartesian coordinate
system changed to the cylindrical that, in which case, the
anisotropy factor for the thickness and the other directions
were put “1”.
The tube was considered as a deformable part and it
was meshed using composite shell elements (four nodes,
reduced integration elements, ABAQUS type S4R).
Friction between the mold and the tube was intended
0.1. The tube was used in the power hardening law to model
its behavior. The Holloman’s equation is written as follows
[17]:
s
Y
¼ Kðe Þ
n
ð14Þ
where
s
Y
is the effective stress, e is the effective plastic
strain, n is strain hardening exponent and K is the strength
coefficient.
Figure 6 demonstrates the FE model included of the
tube and the die.
3.1 Analytical necking criterion
Selecting an appropriate necking criterion is important to
determine the start of plastic instability in tube hydro-
forming. For obtaining the FLC, in this research, necking
criteria, containing the acceleration of maximum and
minimum strain were employed to predict the onset of
plastic instability.
The necking time of a specimen could be determined by
using this method. To obtain the FLC numerically, it was
essential to predict at which time and where the necking
phenomena occurred in the analyzed material. It was
possible to predict the necking time of the analyzed
specimen using its acceleration of the max strain. Two
different criteria to detect the start of plastic instability in
the tube were suggested to determine the FLC. The
forming limits of the tube were predicted, considering the
history of the maximum and minimum strains by taking
the maximum second derivative. For a given strain path,
the limit strain was determined at the maximum value of
the strain acceleration. Figure 7 represents the maximum
and minimum strains for the 4 mm axial feeding mode
(Fig. 3).
Figure 8 shows a relationship between the two criteria
large strain and small strain. Due to the linear relationship
between the two criteria, it is concluded that the second
derivative both at the same time reaches its maximum
value. As a result, the use of either of two criteria will have
one answer.
For this purpose, after completing the simulations, the
element that had the maximum amount of equivalent
plastic strain was reported.
Then, a diagram for the highest and lowest strain versus
time for that element was determined. For example, the
graph for the 4 mm axial feeding mode (Fig. 3) is shown in
Figure 9.
After drawing the curve, get the Microsoft Office Excel
software output from the curve. Then, import that data to
the MATLAB software for using the curve fitting option to
earn chart’s equation and twice derive from it. Figure 10
represents the second derivative of max strain graph and
the data obtained from it (4 mm axial feeding mode) in
MATLAB.
The time when the acceleration of the maximum strain
got its maximum value (0.006 s) was assumed as the start of
necking phenomena in the analyzed material. Finally,
when the second derivative strain reaches its maximum
value, consider the large strain as e
major
and the small strain
as e
minor
.
0
0.05
0.1
0.15
0.2
0.25
-0.15 -0.1 -0.05 0
Major Strain
Minor Strain
FLDs of 7020-T6 tube
Experiment
Simulations
Fig. 11. The FLDs for the aluminum tube 7020-T6.
Table 2. Comparison of the experimental and numerical
major strains for two different strain paths.
Near plane
strain mode
Near uniaxial
tension mode
Experiment 0.05 0.21
FEM 0.03 0.20
A. Afshar et al.: Mechanics & Industry 18, 411 (2017) 5
