Internal state variable plasticity damag
Journal of Materials Processing Technology 210 (2010) 1726–1737
Contents lists available at ScienceDirect
Journal of Materials Processing Technology
journal homepage: www.elsevier.com/locate/jmatprotec
Internal state variable plasticity-damage modeling of the copper
tee-shaped tube hydroforming process
J. Crapps a,∗ , E.B. Marin b , M.F. Horstemeyer a,b , R. Yassar c , P.T. Wang b
a
b
c
Department of Mechanical Engineering, Mississippi State University, Mississippi State, MS, United States
Center for Advanced Vehicular Systems, Mississippi State University, Mississippi State, MS, United States
Department of Mechanical Engineering and Engineering Mechanics Michigan Technological University, Houghton, MI, United States
a r t i c l e
i n f o
Article history:
Received 3 June 2009
Received in revised form 25 May 2010
Accepted 2 June 2010
Keywords:
Internal state variable
Hydroforming
Tee-shaped tube
Finite element
Simulation
Plasticity-fracture
a b s t r a c t
This paper presents a parametric finite element analysis using a history-dependent internal state variable model for a hydroforming process. Experiments were performed for the internal state variable model
correlation and for validating a 2-in. copper tee hydroforming process simulation. The material model
constants were determined from uniaxial stress–strain responses obtained from tensile tests on the tube’s
material. In the finite element simulations, the mesh and boundary conditions were integrated with the
geometry and process parameters currently used in industry. The study provides insights for the variation of different process parameters (velocity and pressure profiles, and bucking system characteristics)
related to the finished product.
© 2010 Elsevier B.V. All rights reserved.
1. Introduction
Tube hydroforming is a metal forming process in which tubes
are formed into complex shapes within a die cavity using internal pressure and axial compressive forces simultaneously. The
development of the initial techniques and establishment of the
theoretical background goes back to the 1940s (Koc and Altan,
2001). Grey et al. (1939) were the first to report on hydroforming of
seamless copper fittings with T-branches. According to Ahmetoglu
et al. (2000), tube hydroforming offers many advantages over
conventional manufacturing methods including: part consolidation, weight reduction, improved structural strength and stiffness,
lower tooling cost due to fewer parts, fewer secondary operations, reduced dimensional variations, and reduced scrap. They also
report that most applications of tube hydroforming can be found
in the auto and aircraft industries as well as manufacturing components for sanitary use.
The hydroforming process has some inherent problems that
include bursting (fracture), wrinkling, and wall thinning, which
strongly depend on the choice of the processing conditions. One
Abbreviations: DHP Copper, Deoxidized High Phosphorus Copper; EMMI, Evolving Microstructural Model of Inelasticity; OFHC, Oxygen Free High Conductivity.
∗ Corresponding author. Tel.: +1 6623123251; fax: +1 6623257223.
E-mail address: [email protected] (J. Crapps).
0924-0136/$ – see front matter © 2010 Elsevier B.V. All rights reserved.
doi:10.1016/j.jmatprotec.2010.06.003
particular hydroforming process in which these inherent problems
often arise is a T-branch type use for fittings. As such the focus
of our study is to analyze a 2-in. copper tee forming process with a
history-dependent internal state variable plasticity model to better
control the process outcome.
Modeling of hydroforming via FEM has been a focal point of
much recent research. Chen et al. (2000) worked to obtain a fundamental understanding of the hydroforming process variables such
as internal pressure, ram movement, and lubricant through corner
fill modeling. Strano et al. (2001) used FEM to develop an adaptive
simulation to select a feasible hydroforming loading path with a
minimum number of runs. The adaptive technique detects the onset
of defects (wrinkling, bursting, buckling) and adjusts the loading
paths accordingly. Shirayori et al. (2002) investigated via FEM and
experiments the free hydraulic bulging of copper and aluminum
alloy tubes. They found that an increase in thickness deviation
during free bulging depended on tube material and end boundary
conditions. Koc (2003) used a finite element simulation to perform
virtual experiments to obtain guidelines on the use of different
loading path schemes. Smith et al. (2004) showed that a strain rate
independent finite element model underestimates the burst pressure for materials featuring higher strain rate sensitivity. Shirayori
et al. (2004) successfully used FEM to design loading paths for
hydroforming processes. Jirathearanat et al. (2004) used finite element simulation to estimate the effects of processing parameters
(pressure level, axial feed, initial tube length) and then to optimize
J. Crapps et al. / Journal of Materials Processing Technology 210 (2010) 1726–1737
process inputs for a forming operation. Chu and Xu (2004) examined the different phenomena of buckling, wrinkling, and bursting
in various aluminum alloys with the goal of obtaining optimal process parameters. Jansson et al. (2005) employed FEM to improve
a hydroforming process and expressed the need for better constitutive models. Abrantes et al. (2005) used FEA to establish a basic
understanding of the tube hydroforming process for aluminum and
copper tubes. Kashani Zadeh and Mashhadi (2006) used finite element simulations to quantify the effects of coefficient of friction,
strain hardening exponent, and fillet radius on the parameters, protrusion height, thickness distribution, and clamping an axial forces.
Guan et al. (2006) developed and implemented a polycrystalline
model into a finite element program and calculated texture evolution on several aluminum hydroformed components. Heo et al.
(2006) used ANSYS parametric design language to design an adaptive FEM simulation for determination of a suitable hydroforming
load path. They validated their model via experiments. Islam et
al. (2006) used finite elements to verify the suitability of hydroforming double layer brass and copper tubes. Their results were
verified experimentally. Daly et al. (2007) recently focused on the
modeling of the post-localization behavior of low carbon steels in
the hydroforming process. Islam et al. (2008) investigated internal
stresses created by hydroforming double layered components. Kim
et al. (2009) constructed a theoretical forming limit stress diagram
to be used with FEM simulations to provide a better method for
simulation-based design of hydroforming processes. (Mohammadi
and Mosavi, 2009) determined the proper loading paths via FEM
and a fuzzy controller.
Modeling of copper tube hydroforming has received some attention as well. Shirayori et al. (2002) investigated the use of a volume
control method for free bulging of copper tube. They found the
volume control method was better for reaching the maximum
hydroforming pressure. Hama et al. (2003) developed a finite element code for hydroforming analysis and compared the hydrostatic
bulge simulation to experiments for a copper tube and found good
agreement. Abrantes et al. (2005) used FEM to establish a basic
understanding of hydroforming of aluminum and copper tubes
with the end goal of finding a process window. Kocanda and
Sadlowska (2006) used FEM simulations and predicted strain localization and bursting via the forming limit curve. They found that
the forming limit curve underestimates the hydroforming limit for
X-joints. Carrado et al. (2008) studied residual stresses in drawn
copper tubes and their relation to geometrical changes in the tube.
The use of advanced physics-based constitutive models for
the behavior of materials during hydroforming is a promising
possibility. Cherouat et al. (2002) used FEM with a thermodynamically coupled constitutive model including thermodynamic
state variables accounting for isotropic hardening and isotropic
ductile damage to investigate the effects of friction coefficient,
material ductility, and hydro bulging condition on the hydroformability of various thin tubes. Varma et al. (2007) used a
anisotropic version of the Gurson model to predict localized necking in an aluminum alloy. They compared their simulation results
with the experiments of Kulkarni et al. (2004). Butcher et al.
(2009) performed computer simulations incorporating a variant of
the Gurson–Tvergaard–Needleman constitutive model to account
for the influence of void shape and shear on coalescence. They
performed a parametric study to determine appropriate void nucleation stress and strain. They found their calibrated model to agree
well with experimentally determined burst pressure. Along these
lines, this study has used the Evolving Microstructural Model of
Inelasticity, EMMI (Marin et al., 2006), to describe the material
response during the hydroforming process. This model has been
formulated in the context of internal state variable theory and couples isotropic plasticity and isotropic damage. The model also has
the ability to capture strain rate and multi-axial stress state effects,
1727
features needed for simulating complex deformation processes. A
brief description of the model is presented in the text.
2. The hydroforming process
A copper blank is formed into a copper tee via the hydroforming process using six components: top and bottom dies, left and
right rams, the bucking system, and a copper blank. The dies are
made of tool steel which is machined to the profile of the desired
copper tee fitting. The rams are also tool steel and are machined to
provide the proper inner radius for the tee fitting. The bucking system comprises a bucking bar, two bucking punches, and a hydraulic
press. The bucking punches are simply solid steel cylinders the size
of the tee branch. The punches fit into the tee branch cavity of the
hydroforming die and inhibit the copper flow into the branch cavity as the tee is forming to prevent process failure due to bursting.
As the bucking punches contact the tee, both the punches and the
bucking bar are lifted until the bucking bar contacts the hydraulic
cylinder, which provides a resistant force additional to the weight
of the bucking bar and bucking punches so that the tee branch will
not form upward too quickly and experience a burst failure.
With the top die raised and the rams retracted, the copper blank
is placed into the bottom die. The top die lowers as the rams move
towards the ends of the copper blank. The ram tips are tapered
and as they enter the copper blank, an interference fit is created. A
mixture of water and a water soluble oil solution called whitewater
is injected into the blank via holes bored through the center of the
rams, thus pressurizing the inside of the blank for hydroforming.
The rams enter the blank until the inside diameter of the blank fits
over the outside diameter of the ram and the end of the blank sits
against the ram shoulder. At this point, a sufficient seal has been
produced for very large pressures to be created inside the blank for
the forming process.
In the next phase of the forming process, the rams compress
the copper blank as the internal pressure, due to the whitewater, is
built gradually to approximately 35–40 MPa. As the rams compress
the blank and the pressure builds, the tee branch forms within the
branch cavity of the top die. As the branch moves upward, it forms
against the bucking system. The bucking system simply exerts a
constant force on the growing tee branch.
After the branch overcomes the bucking force, it continues
growing upward until it hits the hard stop which stops growth at
the desired tee branch height. At a set distance from the hard stop,
the whitewater pressure ramps into what is called coining. During
coining, the whitewater pressure ramps from the forming pressure
of about 35 MPa to approximately 75 MPa. The purpose of coining
is to make sure the radii of the hydroformed tee match the radii of
the forming dies, coining the part, and finishing the process.
The ram movement and the white water pressure are controlled
by time history curves defined by the process control system using
input data from the operator. Fig. 1 presents typical ram velocity
and pressure curves for the hydroforming process. In general, the
quality of the product (tee-shaped copper tube), and hence, the
robustness of the process, depends on the details of these curves.
When the process is not robust, the tube can be ruined. For example, Fig. 2 illustrates the difference between a well-processed tee
with one that incurred wrinkling and one that fractured during the
processing.
3. Simulating the hydroforming process
In order to accurately simulate the complex hydroforming
process described in this paper, three key features need to be
addressed: material model, meshing, and boundary conditions.
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J. Crapps et al. / Journal of Materials Processing Technology 210 (2010) 1726–1737
Fig. 2. Comparison of a well-processed tee-tube, a wrinkled tee-tube, and a fractured tee-tube.
Fig. 1. Typical time history curves for the main processing parameters (ram velocity
and white water pressure) of the hydroforming process.
3.1. Material model
The blank material used in the hydroforming process is
Deoxidized High Phosphorus Copper (DHP Copper) and has a manufacturing history prior to being formed. The material begins in the
form of a cast copper billet which is then extruded into a tube. The
extruded tube then undergoes three draw processes to achieve the
proper tube size for hydroforming. After these deformation steps,
an annealing process is used to increase the ductility (formability)
of the strain-hardened copper tube.
In order to capture the effects of this history to initialize the
material model for the hydroforming process, one must employ
an internal state variable model (Horstemeyer and Wang, 2003).
The material model used to capture the hydroforming process is
the Evolving Microstructural Model of Inelasticity (EMMI) (Marin
et al., 2006). EMMI has a rich history of development in which
many complex, large strain boundary value problems have been
solved starting with the original dislocation based hardening rules
from Bammann (1984, 1990) and Bammann and Aifantis (1987).
Later, damage and failure were added in Bammann et al. (1993)
and Horstemeyer et al. (2000) with the use of the Cocks and
Ashby (1980) void growth rule. Using damage as an internal state
variable helps to keep track of the material’s degradation due
to the evolution of voids or porosity. In general, in finite element simulations, the damage typically increases from its initial
state and could reach 100% within a particular element, which
suggest element failure and a hole in the material. Once this
occurs, the stress in the neighboring elements concentrates and
helps grow the damage in those adjacent elements. Also, because
of different geometric stress concentrations within the boundary value problem, damage will grow in many other elements
to certain levels below the fracture level; however, softening
will occur together with an associated degradation of the elastic
modulus.
In EMMI the equations are formulated in a non-dimensional
form, and are similarly applicable to rate and temperaturedependent deformation of polycrystalline metals. The specific
constitutive relationships of the EMMI model describe the kinematics of large deformation, the elastic and plastic response of the
material as well as damage progression due to void growth, and are
Table 1
Non-dimensional plasticity parameters of EMMI.
a
Functions
and
are expressed as:
where mi , i = 1,5, and
are s and
is a reference temperature. The simulations in this work used:
1982). Other material constants were calibrated to experimental data.
= 0.36, and
= 300 K (Frost and Ashby,
J. Crapps et al. / Journal of Materials Processing Technology 210 (2010) 1726–1737
1729
given by the following equations:
(1)
(2)
(3)
(4)
(5)
expressed in terms of other constants as given in Table 1. Commonly, these parameters are used to fit the predicted stress
response of the model to experimental stress–strain responses
obtained at different temperatures and strain rates. The damage
parameters (m,ϕ0 ), where ϕ0 is the initial void volume fraction,
(6)
(7)
(8)
with
where
denotes a corrotational rate typically used to make
the formulation frame indifferent,
scalar variable,
designates a dimensionless
(bold-faced letter) represents a dimensionless
second-rank tensor, and
indicates a dimensionless time deriva-
tive. In these equations,
are the total stress tensor and
its hydrostatic part (pressure);
tic, and plastic rate of deformation tensors;
are the total, elasare the total
are the internal state variables
and elastic spin tensors;
of the model, the first two representing strengths for kinematic
(tensor) and isotropic (scalar) hardening, and the last one denoting isotropic damage (void volume fraction); is temperature;
is a function describing the thermal expansion characteristics of
the material; and Fv is a function of the Poisson’s ratio. The symbols dev(·), tr(·) and ||(·)|| denote the deviatoric, trace and norm
operators of a second-rank tensor.
The normalized temperature-dependent plasticity parameters
of the model are (
), and they are
are usually obtained by fitting the model response to experimental
load-displacement data obtained from notched tensile specimen
tests. The model has been integrated using a semi-implicit scheme
and implemented in commercial finite element codes such as LSDyna and ABAQUS (for details, see Marin et al., 2006).
This plasticity-damage modeling framework described above
has experienced success in solving a wide variety of large
strain plasticity engineering problems that led to ductile fracture.
Horstemeyer (1992) ran large scale, complex finite element simulations using the model to show the damage progression and
final fracture of submarine hulls. Bammann et al. (1993) used the
model to capture high rate fracture of steel materials with different geometries. Horstemeyer and Revelli (1996) demonstrated
the materials processing history effects on the final fracture of
various high rate conditions. In Horstemeyer (2000) the model
was used for forming limit diagrams of aluminum alloys, which
required the damage model to quantify the localization and fracture that occurred. Horstemeyer et al. (2003) employed the model
to capture the damage progression of magnesium notch specimens
under different stress triaxialities. Fang et al. (2005) employed
the plasticity-damage model to run large scale finite element
simulations of car crashes. Guo et al. (2005) showed how the
plasticity-damage model was used for high rate machining in which
the fracture of the chips was different, and verified by experimen-
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Fig. 3. Experimental and averaged stress–strain responses from copper tensile tests.
Fig. 4. Strain rate effects on the stress–strain response of copper.
tal data, for a variety metal alloys (steel, aluminum, and titanium).
Anurag et al. (2009) applied the model to metal cutting of steel
alloys showing that the model captures the nuances of the different stress states (compression, tension, and torsion) on fracture.
The plasticity-damage model, however, has not been used to model
the hydroforming process until now.
For the present work, the constants for the material model have
been correlated to the experimental stress–strain responses generated from mechanical characterization studies of the copper tube.
Tensile tests were performed on a number of standard dogbone
specimens cut from the annealed copper blank (prior to being
hydroformed). The tests were carried out at room temperature
(25 ◦ C) and at a constant strain rate of 0.001/s. Fig. 3 shows the
stress–strain responses obtained from eight tensile tests. Also presented in Fig. 3 is the calibrated stress–strain curve from EMMI.
Calibration is performed using a matlab code developed for such
purpose (Marin et al., 2006). As the material experiences a range
of strain rates during the hydrofoming process, the material model
should be able to capture the material’s strain rate dependence. One
main parameter of the model that determines the rate sensitivity at
room temperature is the exponent n. In this work, its value has been
estimated from data published by Dao et al. (2007) who reported
a rate sensitivity exponent for copper on the order of 1/n ≈ 0.05.
As such the bounds for n of 20 ± 15% were used in the calibration
routine, giving a computed value of n = c1 + c9 / = 20.7, c9 = 0 (see
Table 2). Fig. 4 presents the model correlations computed from
matlab, the experimental stress–strain response, and published
stress–strain curves from the literature. The varying strain rate copper data shown in Fig. 4 is taken from Tanner et al. (1999) and
Molinari and Ravichandran (2005), where they have used Oxygen
Free High Conductivity (OFHC) copper, a material whose chemical composition is similar to DHP copper. Note the consistency of
our experimental data with the literature data. Finally, the damage
constants are calibrated using the load-displacement curve for a
selected uniaxial tensile test in order to capture the failure strain
during tensile loading. This calibration is performed using finite
elements due to the non-homogeneous deformation of the specimen after maximum loading. Results of this fitting procedure are
presented in Fig. 5. A summary of the EMMI material constants for
DHP copper is given in Table 2. Note that most of the temperaturerelated constants have been set to zero as the experimental data
available is limited to one testing temperature (room temperature).
During hydroforming, a material point is typically subjected to
multi-axial stress states but a biaxial state of stress usually predominates. Hence, hydroforming has been analyzed in the context
of tube bulge tests. Fuchizawa and Narazaki (1993) was one of the
first to show that a biaxial stress state was more compliant than
the response under uniaxial loading. Koc et al. (2001) developed
a methodology which can be used to obtain the flow stress curve
of tubular materials using on line automated hydraulic bulge tests.
Strano et al. (2004) developed an inverse energy approach to determine the flow stress of tubular materials from the bulge test based
Table 2
DHP copper material constants of EMMI.
⌣
c1
⌣
c2
⌣
c3
⌣
c4
⌣
c5
⌣
c6
⌣
c7
⌣
c8
⌣
c9
⌣
c 10
⌣
m
Plasticity material constants
20.71
1.36 × 10−6
5.48 × 10+2
7.8617 × 10−3
2.82 × 10+1
2.0 × 10−2
0.0
2.9667 × 10−4
0.0
1.0
Damage material constants
4.5
⌣
Q1
⌣
Q2
⌣
Q3
⌣
Q4
⌣
Q5
⌣
m1
⌣
m2
⌣
m3
⌣
m4
⌣
m5
ϕ0
0.0
0.0
0.0
0.0
0.0
1.21
2.70
1.08
8.46
0.547
10−4
Fig. 5. Fitting the damage parameters (ϕ0 = Phio , m = DMR) of EMMI: (a) finite element mesh of uniaxial tensile specimen, and (b) fitted load-displacement curve for
a number of values of m. Best fit is obtained with: ϕ0 = 10−4 , m = 4.5.
J. Crapps et al. / Journal of Materials Processing Technology 210 (2010) 1726–1737
1731
on an energy balance. Hwang and Lin (2002) developed an analytical relationship investigating the relationship between bulge
height and internal pressure for bulge forming. Sokolowski et al.
(2000) coupled a hydraulic bulging tool set with Finite Element
simulations to determine the proper flow stress of tubular materials. Horstemeyer (2000) showed that these multi-axial stress states
can be captured by the proposed internal state variable modeling
methodology as demonstrated by forming limit diagrams.
3.2. Finite element mesh
The hydroforming process was carried out using six separate
components but the mesh comprised only four parts as illustrated
in Fig. 6: die, bucking plate, ram, and copper billet. The entire finite
element model contains 3349 elements chosen from the ABAQUS
element library and distributed as follows: 1536 C3D8R (brick) elements for the copper billet, 21 ARSR rigid elements for the buckling
plate, 330 R3D4 elements for the ram, and 1462 R3D4 elements for
the die. The deformation of the blank was symmetric along two
axes, so a quarter model mesh was implemented to speed up computational times. Fig. 7 shows that this quarter model only involves
one ram, a quarter of the blank, and top and bottom dies. Part geometry for the hydroforming dies was imported into ABAQUS CAE as
a rigid solid. The hydroforming rams were modeled in ABAQUS as
cylindrical rigid shells. The bucking system was modeled as a rigid
plate the size of the tee branch cavity in the dies. The dies, rams
and bucking system were assumed to be rigid, because of their tool
steel composition, which is much harder/stronger than the copper
blank.
Fig. 6. The finite element simulation set-up showing the different components of
hydroforming process.
3.3. Boundary conditions
The hydroforming process was controlled by a ram
velocity–time history and a pressure–time history curve (see
Fig. 1) and was broken into four steps for the finite element
simulations. The pressure–time history controlled the magnitude
of the whitewater pressure within the copper blank during the
hydroforming process. A more detailed description of the pressureand velocity–time histories used for the process is shown in Fig. 8
along with the finite element steps employed for the boundary
Fig. 7. Hydroforming simulation steps and bucking plate boundary conditions.
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J. Crapps et al. / Journal of Materials Processing Technology 210 (2010) 1726–1737
Fig. 8. Simulation steps on pressure–time and velocity–time histories.
conditions (see Fig. 7). The pressure–time history comprised three
stages: sealing, forming, and coining. The ram velocity–time history comprised two stages: sealing and forming. The sealing stage
for both time histories coincided and the ram forming portion
overlapped the forming and coining portions of the pressure–time
histories. During the sealing stage, the ram entered into the tube
to seal the end of the copper blank. Once the ram sealed the copper
blank, the pressure was initiated to reach the set sealing pressure.
During the forming stage, the ram moved slower as the pressure
grew to the full forming pressure and the branch of the tee formed.
The tee branch formed upward until it pressed the bucking plate
to within 0.010 in. (.254 mm) of the hard stop. At this time, the
pressure entered the coining stage, ramping to approximately
75 MPa and causing the tee branch to form into all the corners of
the die.
The pressure–time history was generated based upon six
parameters: maximum sealing pressure, sealing pressure ramp
time, maximum forming pressure, forming pressure ramp time,
coining pressure, and coining pressure ramp time. The pressure
increased from its current value to the set value over the ramp time
by following an S-curve. The S-curve was created by fitting a sine
wave over the acceleration and deceleration portions of a constant
acceleration trapezoidal curve, as shown in Fig. 8. Once the pressure curve ramped to the desired pressure, a constant pressure was
maintained until the next stage of the forming process was reached.
The ram velocity–time history displayed in Fig. 8 was generated based on six parameters: sealing acceleration distance, sealing
maximum velocity, sealing travel, forming acceleration distance,
forming maximum velocity, and forming travel. The ram velocity increased from zero to the specified velocity over the specified
Fig. 9. Snapshots of the von Mises stress history are shown as a function of time. The inset defines the pressure and velocity for each snapshot in time.
J. Crapps et al. / Journal of Materials Processing Technology 210 (2010) 1726–1737
1733
Fig. 10. The effective plastic strain is shown in the color contour for different simulations with the forming pressure decreasing for each simulation. The inset
pressure–time history plots illustrate the different simulations with six different pressure histories. Clearly, the higher the gradient between 2 and 4 s yields greater plastic
strains.
acceleration distance by following an S-curve similar to the pressure curve. The ram S-curve was also a sine wave fit over a constant
acceleration trapezoidal curve. Once the ram velocity curve accelerated to the specified velocity, a constant velocity was maintained
until the deceleration portion of the curve was reached. The total
ram travel distance was equal to the area under the ram velocity curve. The control system (physical and simulated) calculated
the area under the velocity–time history curve. If this area was
less than the total travel distance, a constant velocity was added
to the ram velocity–time history to reach the specified travel distance. If this area was more than the total travel distance, the
acceleration distance was reassigned to half the total travel distance and the maximum velocity was reassigned to the velocity
which would have been reached using the original acceleration
distance.
Now that the ram movements have been discussed, assumptions related to the other components will be explained. The die
was fixed in such a manner that no displacements or rotations were
admitted in any direction. The bucking plate was force controlled
as it was designed to inhibit the branch of the tee so fracture would
not occur. The bucking plate was able to move as close as 1.524 mm
from the outer diameter of the copper billet. Master-slave surface
contacts existed between the copper blank and the die, ram, and
bucking plate.
3.4. Preliminary simulation results
Fig. 9 shows snapshots at different times of the von Mises stress
in the deforming copper blank during the hydroforming simulation.
These results were obtained using EMMI without damage (ϕ0 = 0).
These snapshots depict the connection between the pressure- and
velocity–time histories with the stress contours and deforming
geometry of the tube throughout the forming process. Clearly, the
developed finite element model and corresponding solution procedure with four different steps (see Fig. 7) made the numerical
simulation very robust, capturing well the details of the hydroforming process.
To better understand the effect of the white water pressure on
the formation of the tee branch, a number of simulations were run
where the transition time from the sealing pressure to the forming
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J. Crapps et al. / Journal of Materials Processing Technology 210 (2010) 1726–1737
Fig. 11. The damage level is shown in the color contour for different simulations with the forming pressure decreasing for each simulation. The inset pressure–time
history plots illustrate the different simulations with six different pressure histories. Clearly, the higher pressure gradient between 2 and 4 s yields greater damage. Note
the apparent volume increase of the elements with high levels of damage (porosity). High values of damage strongly affect the deformation patterns of the hydroformed
tube.
pressure (pressure gradient) was varied from approximately 2 to
6 s. Again, this study was performed with EMMI without damage.
The results from these simulations are presented in Fig. 10, where
contour plots of the effective plastic strains at the end of the process are given. Two observations are important from this figure.
First, extremely large strains occur when the pressure gradient is
high. Simulations 1–3 all show effective strains up to 500%. Second, wrinkling (instability from localized buckling) starts to occur
as the forming pressure is decreased. Simulation 6 shows the wrinkling starting in the radius of the tee. On the other hand, simulation
4 and 5 indicates that the pressure gradients used for these cases
are adequate to form the tee branch, although simulation 5 uses
a lower forming pressure. This numerical study illustrates the fact
that there exists a window of optimum processing parameters that
produces well-processed tee-tubes.
Another group of simulations with identical boundary conditions as those described above was performed using the EMMI
model with damage. The initial value of damage was set to
ϕ0 = 0.001. The results from these simulations, given in Fig. 11,
are presented in terms of contour plot of damage (void volume
fraction) at the end of the forming process. As shown by this figure, the pressure gradient has a strong effect on the evolution
of damage, with the model predicting porosity levels of up to
one where the pressure transients are short (simulations 1–4).
Note that this may indicate bursting or fracture at the top of the
copper tee. Also note that damage levels that induced fracture
J. Crapps et al. / Journal of Materials Processing Technology 210 (2010) 1726–1737
1735
Fig. 13. Comparison of finite element simulation results with the experimental
results from a processed hydroformed tee-tube that induced wrinkling.
Fig. 12. Comparison of the finite element simulation results with the experimental
results from a well-processed hydroformed tee-tube.
align with the extremely large strains exhibited in Fig. 10. It is
important to point out here that, as the hydroforming process is
a pressure-driven process, one may expect that the tube blank will
experience high stress triaxialities (ratio of the hydrostatic stress
to von Mises equivalent stress) in zones with high tensile hydrostatic stresses. In fact, the simulations showed maximum stress
triaxiality values ranging from 1.3 to 6, depending on the boundary conditions. Since the void growth rule used by EMMI depends
exponentially on this quantity, high void growth rates are expected
to happen for hydroforming boundary conditions inducing large
tensile stresses. This behavior has been observed in the numerical
simulations.
The above simulations evaluated the robustness of the finite element model constructed as well as the general capability of the
EMMI model to predict the large plastic deformations and ductile
damage evolution during the hydroforming process. As validation
experiments for the damage aspects of the hydrofoming process
is lacking (effect of processing parameters on the failure/fracture
of copper tee-tubes), the next section of the paper mainly
focuses on validating both the process model and the plasticity
aspects of EMMI with specific experimental data collected during
the study.
4. Simulation/validation results and discussion
Once the aforementioned hydroforming simulations were performed, an experimental validation study was conducted at a
hydroforming plant. One check was to compare the material
flow patterns of points in the copper blank with that obtained
from the numerical simulations. In the plant, the forming process was set by five main parameters: ram forming travel, forming
Table 3
Validation experiments for hydroforming simulation.
Test
Ram travel (mm)
Pressure (MPa)
Form P ramp (s)
Buck position (mm)
Buck force (N)a
Nominal
Buck force (stunt)
38.862
38.862
38.862
38.862
38.862
34.474
34.474
34.474
34.474
34.474
5.5
5.5
5.5
5.5
5.5
10.16
10.16
10.16
10.16
10.16
1001
1112
1223
1334
278
Buck force (blow)
a
Force values are for a quarter hydroforming model.
1736
J. Crapps et al. / Journal of Materials Processing Technology 210 (2010) 1726–1737
Fig. 14. Comparison of material point profiles for the hydroforming validation
experiments and the associated numerical simulation results with a varying friction coefficient. Note that when the coefficient of friction was = 0.075 the results
compared extremely well.
Fig. 15. Comparison of material point profiles for the hydroforming validation
experiments and the associated numerical simulation results with a bucking resistance. Note that all cases compare very well.
5. Conclusions
pressure, forming pressure ramp time, bucking punch position,
and bucking punch force. For the validation study, some of
these parameters were varied in the physical process as presented by Table 3. Fig. 12 shows clearly that the numerical
predictions compared well with the experimental results for a
well-processed tee-tube. Fig. 13 also shows a great comparison
of the simulation results with the experiment for a wrinkled
tee-tube.
For the simulations related to each set of conditions presented
in Table 3, the physical profile of the tubes were measured and
compared to the simulation profiles. A simulation of the nominal case was used to estimate the friction coefficient for the
hydroforming process using a methodology similar to Ngaile et
al. (2004). Fig. 14 presents the physical and simulation profiles of
the overall tee and the bottom section of the tee for the nominal
case. The comparison shows that the friction coefficient providing
the best behavior had a value of = 0.075. This value corroborates results found in Ngaile et al. (2004). The small difference
in the bottom wall thickness is probably due to alignment and
tolerance issues in the hydroforming tooling. Using the coefficient of friction of 0.075, Fig. 15 shows that the overall and
bottom profile for each case included in the validation experiments agrees well with those obtained from the hydroforming
experiments.
A modeling-experimental study was performed on a complex
hydroforming process using a non-dimensionalized internal state
variable history model (EMMI) that captures strain rate and multiaxial stress state effects. The model constants were determined
from uniaxial stress–strain responses and the predictive model was
used to evaluate different processing histories. The model was validated from experimental results obtained from in-plant processes.
The excellent comparisons indicate that the model can be used to
optimize the process space to eliminate wrinkling or fracture. The
developed model is used in a second article to perform a parametric
study of the copper T-shape tube hydroforming process.
Acknowledgement
This work was performed under the auspices of the Center for
Advanced Vehicular Systems (CAVS) at Missisissippi State University.
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Contents lists available at ScienceDirect
Journal of Materials Processing Technology
journal homepage: www.elsevier.com/locate/jmatprotec
Internal state variable plasticity-damage modeling of the copper
tee-shaped tube hydroforming process
J. Crapps a,∗ , E.B. Marin b , M.F. Horstemeyer a,b , R. Yassar c , P.T. Wang b
a
b
c
Department of Mechanical Engineering, Mississippi State University, Mississippi State, MS, United States
Center for Advanced Vehicular Systems, Mississippi State University, Mississippi State, MS, United States
Department of Mechanical Engineering and Engineering Mechanics Michigan Technological University, Houghton, MI, United States
a r t i c l e
i n f o
Article history:
Received 3 June 2009
Received in revised form 25 May 2010
Accepted 2 June 2010
Keywords:
Internal state variable
Hydroforming
Tee-shaped tube
Finite element
Simulation
Plasticity-fracture
a b s t r a c t
This paper presents a parametric finite element analysis using a history-dependent internal state variable model for a hydroforming process. Experiments were performed for the internal state variable model
correlation and for validating a 2-in. copper tee hydroforming process simulation. The material model
constants were determined from uniaxial stress–strain responses obtained from tensile tests on the tube’s
material. In the finite element simulations, the mesh and boundary conditions were integrated with the
geometry and process parameters currently used in industry. The study provides insights for the variation of different process parameters (velocity and pressure profiles, and bucking system characteristics)
related to the finished product.
© 2010 Elsevier B.V. All rights reserved.
1. Introduction
Tube hydroforming is a metal forming process in which tubes
are formed into complex shapes within a die cavity using internal pressure and axial compressive forces simultaneously. The
development of the initial techniques and establishment of the
theoretical background goes back to the 1940s (Koc and Altan,
2001). Grey et al. (1939) were the first to report on hydroforming of
seamless copper fittings with T-branches. According to Ahmetoglu
et al. (2000), tube hydroforming offers many advantages over
conventional manufacturing methods including: part consolidation, weight reduction, improved structural strength and stiffness,
lower tooling cost due to fewer parts, fewer secondary operations, reduced dimensional variations, and reduced scrap. They also
report that most applications of tube hydroforming can be found
in the auto and aircraft industries as well as manufacturing components for sanitary use.
The hydroforming process has some inherent problems that
include bursting (fracture), wrinkling, and wall thinning, which
strongly depend on the choice of the processing conditions. One
Abbreviations: DHP Copper, Deoxidized High Phosphorus Copper; EMMI, Evolving Microstructural Model of Inelasticity; OFHC, Oxygen Free High Conductivity.
∗ Corresponding author. Tel.: +1 6623123251; fax: +1 6623257223.
E-mail address: [email protected] (J. Crapps).
0924-0136/$ – see front matter © 2010 Elsevier B.V. All rights reserved.
doi:10.1016/j.jmatprotec.2010.06.003
particular hydroforming process in which these inherent problems
often arise is a T-branch type use for fittings. As such the focus
of our study is to analyze a 2-in. copper tee forming process with a
history-dependent internal state variable plasticity model to better
control the process outcome.
Modeling of hydroforming via FEM has been a focal point of
much recent research. Chen et al. (2000) worked to obtain a fundamental understanding of the hydroforming process variables such
as internal pressure, ram movement, and lubricant through corner
fill modeling. Strano et al. (2001) used FEM to develop an adaptive
simulation to select a feasible hydroforming loading path with a
minimum number of runs. The adaptive technique detects the onset
of defects (wrinkling, bursting, buckling) and adjusts the loading
paths accordingly. Shirayori et al. (2002) investigated via FEM and
experiments the free hydraulic bulging of copper and aluminum
alloy tubes. They found that an increase in thickness deviation
during free bulging depended on tube material and end boundary
conditions. Koc (2003) used a finite element simulation to perform
virtual experiments to obtain guidelines on the use of different
loading path schemes. Smith et al. (2004) showed that a strain rate
independent finite element model underestimates the burst pressure for materials featuring higher strain rate sensitivity. Shirayori
et al. (2004) successfully used FEM to design loading paths for
hydroforming processes. Jirathearanat et al. (2004) used finite element simulation to estimate the effects of processing parameters
(pressure level, axial feed, initial tube length) and then to optimize
J. Crapps et al. / Journal of Materials Processing Technology 210 (2010) 1726–1737
process inputs for a forming operation. Chu and Xu (2004) examined the different phenomena of buckling, wrinkling, and bursting
in various aluminum alloys with the goal of obtaining optimal process parameters. Jansson et al. (2005) employed FEM to improve
a hydroforming process and expressed the need for better constitutive models. Abrantes et al. (2005) used FEA to establish a basic
understanding of the tube hydroforming process for aluminum and
copper tubes. Kashani Zadeh and Mashhadi (2006) used finite element simulations to quantify the effects of coefficient of friction,
strain hardening exponent, and fillet radius on the parameters, protrusion height, thickness distribution, and clamping an axial forces.
Guan et al. (2006) developed and implemented a polycrystalline
model into a finite element program and calculated texture evolution on several aluminum hydroformed components. Heo et al.
(2006) used ANSYS parametric design language to design an adaptive FEM simulation for determination of a suitable hydroforming
load path. They validated their model via experiments. Islam et
al. (2006) used finite elements to verify the suitability of hydroforming double layer brass and copper tubes. Their results were
verified experimentally. Daly et al. (2007) recently focused on the
modeling of the post-localization behavior of low carbon steels in
the hydroforming process. Islam et al. (2008) investigated internal
stresses created by hydroforming double layered components. Kim
et al. (2009) constructed a theoretical forming limit stress diagram
to be used with FEM simulations to provide a better method for
simulation-based design of hydroforming processes. (Mohammadi
and Mosavi, 2009) determined the proper loading paths via FEM
and a fuzzy controller.
Modeling of copper tube hydroforming has received some attention as well. Shirayori et al. (2002) investigated the use of a volume
control method for free bulging of copper tube. They found the
volume control method was better for reaching the maximum
hydroforming pressure. Hama et al. (2003) developed a finite element code for hydroforming analysis and compared the hydrostatic
bulge simulation to experiments for a copper tube and found good
agreement. Abrantes et al. (2005) used FEM to establish a basic
understanding of hydroforming of aluminum and copper tubes
with the end goal of finding a process window. Kocanda and
Sadlowska (2006) used FEM simulations and predicted strain localization and bursting via the forming limit curve. They found that
the forming limit curve underestimates the hydroforming limit for
X-joints. Carrado et al. (2008) studied residual stresses in drawn
copper tubes and their relation to geometrical changes in the tube.
The use of advanced physics-based constitutive models for
the behavior of materials during hydroforming is a promising
possibility. Cherouat et al. (2002) used FEM with a thermodynamically coupled constitutive model including thermodynamic
state variables accounting for isotropic hardening and isotropic
ductile damage to investigate the effects of friction coefficient,
material ductility, and hydro bulging condition on the hydroformability of various thin tubes. Varma et al. (2007) used a
anisotropic version of the Gurson model to predict localized necking in an aluminum alloy. They compared their simulation results
with the experiments of Kulkarni et al. (2004). Butcher et al.
(2009) performed computer simulations incorporating a variant of
the Gurson–Tvergaard–Needleman constitutive model to account
for the influence of void shape and shear on coalescence. They
performed a parametric study to determine appropriate void nucleation stress and strain. They found their calibrated model to agree
well with experimentally determined burst pressure. Along these
lines, this study has used the Evolving Microstructural Model of
Inelasticity, EMMI (Marin et al., 2006), to describe the material
response during the hydroforming process. This model has been
formulated in the context of internal state variable theory and couples isotropic plasticity and isotropic damage. The model also has
the ability to capture strain rate and multi-axial stress state effects,
1727
features needed for simulating complex deformation processes. A
brief description of the model is presented in the text.
2. The hydroforming process
A copper blank is formed into a copper tee via the hydroforming process using six components: top and bottom dies, left and
right rams, the bucking system, and a copper blank. The dies are
made of tool steel which is machined to the profile of the desired
copper tee fitting. The rams are also tool steel and are machined to
provide the proper inner radius for the tee fitting. The bucking system comprises a bucking bar, two bucking punches, and a hydraulic
press. The bucking punches are simply solid steel cylinders the size
of the tee branch. The punches fit into the tee branch cavity of the
hydroforming die and inhibit the copper flow into the branch cavity as the tee is forming to prevent process failure due to bursting.
As the bucking punches contact the tee, both the punches and the
bucking bar are lifted until the bucking bar contacts the hydraulic
cylinder, which provides a resistant force additional to the weight
of the bucking bar and bucking punches so that the tee branch will
not form upward too quickly and experience a burst failure.
With the top die raised and the rams retracted, the copper blank
is placed into the bottom die. The top die lowers as the rams move
towards the ends of the copper blank. The ram tips are tapered
and as they enter the copper blank, an interference fit is created. A
mixture of water and a water soluble oil solution called whitewater
is injected into the blank via holes bored through the center of the
rams, thus pressurizing the inside of the blank for hydroforming.
The rams enter the blank until the inside diameter of the blank fits
over the outside diameter of the ram and the end of the blank sits
against the ram shoulder. At this point, a sufficient seal has been
produced for very large pressures to be created inside the blank for
the forming process.
In the next phase of the forming process, the rams compress
the copper blank as the internal pressure, due to the whitewater, is
built gradually to approximately 35–40 MPa. As the rams compress
the blank and the pressure builds, the tee branch forms within the
branch cavity of the top die. As the branch moves upward, it forms
against the bucking system. The bucking system simply exerts a
constant force on the growing tee branch.
After the branch overcomes the bucking force, it continues
growing upward until it hits the hard stop which stops growth at
the desired tee branch height. At a set distance from the hard stop,
the whitewater pressure ramps into what is called coining. During
coining, the whitewater pressure ramps from the forming pressure
of about 35 MPa to approximately 75 MPa. The purpose of coining
is to make sure the radii of the hydroformed tee match the radii of
the forming dies, coining the part, and finishing the process.
The ram movement and the white water pressure are controlled
by time history curves defined by the process control system using
input data from the operator. Fig. 1 presents typical ram velocity
and pressure curves for the hydroforming process. In general, the
quality of the product (tee-shaped copper tube), and hence, the
robustness of the process, depends on the details of these curves.
When the process is not robust, the tube can be ruined. For example, Fig. 2 illustrates the difference between a well-processed tee
with one that incurred wrinkling and one that fractured during the
processing.
3. Simulating the hydroforming process
In order to accurately simulate the complex hydroforming
process described in this paper, three key features need to be
addressed: material model, meshing, and boundary conditions.
1728
J. Crapps et al. / Journal of Materials Processing Technology 210 (2010) 1726–1737
Fig. 2. Comparison of a well-processed tee-tube, a wrinkled tee-tube, and a fractured tee-tube.
Fig. 1. Typical time history curves for the main processing parameters (ram velocity
and white water pressure) of the hydroforming process.
3.1. Material model
The blank material used in the hydroforming process is
Deoxidized High Phosphorus Copper (DHP Copper) and has a manufacturing history prior to being formed. The material begins in the
form of a cast copper billet which is then extruded into a tube. The
extruded tube then undergoes three draw processes to achieve the
proper tube size for hydroforming. After these deformation steps,
an annealing process is used to increase the ductility (formability)
of the strain-hardened copper tube.
In order to capture the effects of this history to initialize the
material model for the hydroforming process, one must employ
an internal state variable model (Horstemeyer and Wang, 2003).
The material model used to capture the hydroforming process is
the Evolving Microstructural Model of Inelasticity (EMMI) (Marin
et al., 2006). EMMI has a rich history of development in which
many complex, large strain boundary value problems have been
solved starting with the original dislocation based hardening rules
from Bammann (1984, 1990) and Bammann and Aifantis (1987).
Later, damage and failure were added in Bammann et al. (1993)
and Horstemeyer et al. (2000) with the use of the Cocks and
Ashby (1980) void growth rule. Using damage as an internal state
variable helps to keep track of the material’s degradation due
to the evolution of voids or porosity. In general, in finite element simulations, the damage typically increases from its initial
state and could reach 100% within a particular element, which
suggest element failure and a hole in the material. Once this
occurs, the stress in the neighboring elements concentrates and
helps grow the damage in those adjacent elements. Also, because
of different geometric stress concentrations within the boundary value problem, damage will grow in many other elements
to certain levels below the fracture level; however, softening
will occur together with an associated degradation of the elastic
modulus.
In EMMI the equations are formulated in a non-dimensional
form, and are similarly applicable to rate and temperaturedependent deformation of polycrystalline metals. The specific
constitutive relationships of the EMMI model describe the kinematics of large deformation, the elastic and plastic response of the
material as well as damage progression due to void growth, and are
Table 1
Non-dimensional plasticity parameters of EMMI.
a
Functions
and
are expressed as:
where mi , i = 1,5, and
are s and
is a reference temperature. The simulations in this work used:
1982). Other material constants were calibrated to experimental data.
= 0.36, and
= 300 K (Frost and Ashby,
J. Crapps et al. / Journal of Materials Processing Technology 210 (2010) 1726–1737
1729
given by the following equations:
(1)
(2)
(3)
(4)
(5)
expressed in terms of other constants as given in Table 1. Commonly, these parameters are used to fit the predicted stress
response of the model to experimental stress–strain responses
obtained at different temperatures and strain rates. The damage
parameters (m,ϕ0 ), where ϕ0 is the initial void volume fraction,
(6)
(7)
(8)
with
where
denotes a corrotational rate typically used to make
the formulation frame indifferent,
scalar variable,
designates a dimensionless
(bold-faced letter) represents a dimensionless
second-rank tensor, and
indicates a dimensionless time deriva-
tive. In these equations,
are the total stress tensor and
its hydrostatic part (pressure);
tic, and plastic rate of deformation tensors;
are the total, elasare the total
are the internal state variables
and elastic spin tensors;
of the model, the first two representing strengths for kinematic
(tensor) and isotropic (scalar) hardening, and the last one denoting isotropic damage (void volume fraction); is temperature;
is a function describing the thermal expansion characteristics of
the material; and Fv is a function of the Poisson’s ratio. The symbols dev(·), tr(·) and ||(·)|| denote the deviatoric, trace and norm
operators of a second-rank tensor.
The normalized temperature-dependent plasticity parameters
of the model are (
), and they are
are usually obtained by fitting the model response to experimental
load-displacement data obtained from notched tensile specimen
tests. The model has been integrated using a semi-implicit scheme
and implemented in commercial finite element codes such as LSDyna and ABAQUS (for details, see Marin et al., 2006).
This plasticity-damage modeling framework described above
has experienced success in solving a wide variety of large
strain plasticity engineering problems that led to ductile fracture.
Horstemeyer (1992) ran large scale, complex finite element simulations using the model to show the damage progression and
final fracture of submarine hulls. Bammann et al. (1993) used the
model to capture high rate fracture of steel materials with different geometries. Horstemeyer and Revelli (1996) demonstrated
the materials processing history effects on the final fracture of
various high rate conditions. In Horstemeyer (2000) the model
was used for forming limit diagrams of aluminum alloys, which
required the damage model to quantify the localization and fracture that occurred. Horstemeyer et al. (2003) employed the model
to capture the damage progression of magnesium notch specimens
under different stress triaxialities. Fang et al. (2005) employed
the plasticity-damage model to run large scale finite element
simulations of car crashes. Guo et al. (2005) showed how the
plasticity-damage model was used for high rate machining in which
the fracture of the chips was different, and verified by experimen-
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J. Crapps et al. / Journal of Materials Processing Technology 210 (2010) 1726–1737
Fig. 3. Experimental and averaged stress–strain responses from copper tensile tests.
Fig. 4. Strain rate effects on the stress–strain response of copper.
tal data, for a variety metal alloys (steel, aluminum, and titanium).
Anurag et al. (2009) applied the model to metal cutting of steel
alloys showing that the model captures the nuances of the different stress states (compression, tension, and torsion) on fracture.
The plasticity-damage model, however, has not been used to model
the hydroforming process until now.
For the present work, the constants for the material model have
been correlated to the experimental stress–strain responses generated from mechanical characterization studies of the copper tube.
Tensile tests were performed on a number of standard dogbone
specimens cut from the annealed copper blank (prior to being
hydroformed). The tests were carried out at room temperature
(25 ◦ C) and at a constant strain rate of 0.001/s. Fig. 3 shows the
stress–strain responses obtained from eight tensile tests. Also presented in Fig. 3 is the calibrated stress–strain curve from EMMI.
Calibration is performed using a matlab code developed for such
purpose (Marin et al., 2006). As the material experiences a range
of strain rates during the hydrofoming process, the material model
should be able to capture the material’s strain rate dependence. One
main parameter of the model that determines the rate sensitivity at
room temperature is the exponent n. In this work, its value has been
estimated from data published by Dao et al. (2007) who reported
a rate sensitivity exponent for copper on the order of 1/n ≈ 0.05.
As such the bounds for n of 20 ± 15% were used in the calibration
routine, giving a computed value of n = c1 + c9 / = 20.7, c9 = 0 (see
Table 2). Fig. 4 presents the model correlations computed from
matlab, the experimental stress–strain response, and published
stress–strain curves from the literature. The varying strain rate copper data shown in Fig. 4 is taken from Tanner et al. (1999) and
Molinari and Ravichandran (2005), where they have used Oxygen
Free High Conductivity (OFHC) copper, a material whose chemical composition is similar to DHP copper. Note the consistency of
our experimental data with the literature data. Finally, the damage
constants are calibrated using the load-displacement curve for a
selected uniaxial tensile test in order to capture the failure strain
during tensile loading. This calibration is performed using finite
elements due to the non-homogeneous deformation of the specimen after maximum loading. Results of this fitting procedure are
presented in Fig. 5. A summary of the EMMI material constants for
DHP copper is given in Table 2. Note that most of the temperaturerelated constants have been set to zero as the experimental data
available is limited to one testing temperature (room temperature).
During hydroforming, a material point is typically subjected to
multi-axial stress states but a biaxial state of stress usually predominates. Hence, hydroforming has been analyzed in the context
of tube bulge tests. Fuchizawa and Narazaki (1993) was one of the
first to show that a biaxial stress state was more compliant than
the response under uniaxial loading. Koc et al. (2001) developed
a methodology which can be used to obtain the flow stress curve
of tubular materials using on line automated hydraulic bulge tests.
Strano et al. (2004) developed an inverse energy approach to determine the flow stress of tubular materials from the bulge test based
Table 2
DHP copper material constants of EMMI.
⌣
c1
⌣
c2
⌣
c3
⌣
c4
⌣
c5
⌣
c6
⌣
c7
⌣
c8
⌣
c9
⌣
c 10
⌣
m
Plasticity material constants
20.71
1.36 × 10−6
5.48 × 10+2
7.8617 × 10−3
2.82 × 10+1
2.0 × 10−2
0.0
2.9667 × 10−4
0.0
1.0
Damage material constants
4.5
⌣
Q1
⌣
Q2
⌣
Q3
⌣
Q4
⌣
Q5
⌣
m1
⌣
m2
⌣
m3
⌣
m4
⌣
m5
ϕ0
0.0
0.0
0.0
0.0
0.0
1.21
2.70
1.08
8.46
0.547
10−4
Fig. 5. Fitting the damage parameters (ϕ0 = Phio , m = DMR) of EMMI: (a) finite element mesh of uniaxial tensile specimen, and (b) fitted load-displacement curve for
a number of values of m. Best fit is obtained with: ϕ0 = 10−4 , m = 4.5.
J. Crapps et al. / Journal of Materials Processing Technology 210 (2010) 1726–1737
1731
on an energy balance. Hwang and Lin (2002) developed an analytical relationship investigating the relationship between bulge
height and internal pressure for bulge forming. Sokolowski et al.
(2000) coupled a hydraulic bulging tool set with Finite Element
simulations to determine the proper flow stress of tubular materials. Horstemeyer (2000) showed that these multi-axial stress states
can be captured by the proposed internal state variable modeling
methodology as demonstrated by forming limit diagrams.
3.2. Finite element mesh
The hydroforming process was carried out using six separate
components but the mesh comprised only four parts as illustrated
in Fig. 6: die, bucking plate, ram, and copper billet. The entire finite
element model contains 3349 elements chosen from the ABAQUS
element library and distributed as follows: 1536 C3D8R (brick) elements for the copper billet, 21 ARSR rigid elements for the buckling
plate, 330 R3D4 elements for the ram, and 1462 R3D4 elements for
the die. The deformation of the blank was symmetric along two
axes, so a quarter model mesh was implemented to speed up computational times. Fig. 7 shows that this quarter model only involves
one ram, a quarter of the blank, and top and bottom dies. Part geometry for the hydroforming dies was imported into ABAQUS CAE as
a rigid solid. The hydroforming rams were modeled in ABAQUS as
cylindrical rigid shells. The bucking system was modeled as a rigid
plate the size of the tee branch cavity in the dies. The dies, rams
and bucking system were assumed to be rigid, because of their tool
steel composition, which is much harder/stronger than the copper
blank.
Fig. 6. The finite element simulation set-up showing the different components of
hydroforming process.
3.3. Boundary conditions
The hydroforming process was controlled by a ram
velocity–time history and a pressure–time history curve (see
Fig. 1) and was broken into four steps for the finite element
simulations. The pressure–time history controlled the magnitude
of the whitewater pressure within the copper blank during the
hydroforming process. A more detailed description of the pressureand velocity–time histories used for the process is shown in Fig. 8
along with the finite element steps employed for the boundary
Fig. 7. Hydroforming simulation steps and bucking plate boundary conditions.
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J. Crapps et al. / Journal of Materials Processing Technology 210 (2010) 1726–1737
Fig. 8. Simulation steps on pressure–time and velocity–time histories.
conditions (see Fig. 7). The pressure–time history comprised three
stages: sealing, forming, and coining. The ram velocity–time history comprised two stages: sealing and forming. The sealing stage
for both time histories coincided and the ram forming portion
overlapped the forming and coining portions of the pressure–time
histories. During the sealing stage, the ram entered into the tube
to seal the end of the copper blank. Once the ram sealed the copper
blank, the pressure was initiated to reach the set sealing pressure.
During the forming stage, the ram moved slower as the pressure
grew to the full forming pressure and the branch of the tee formed.
The tee branch formed upward until it pressed the bucking plate
to within 0.010 in. (.254 mm) of the hard stop. At this time, the
pressure entered the coining stage, ramping to approximately
75 MPa and causing the tee branch to form into all the corners of
the die.
The pressure–time history was generated based upon six
parameters: maximum sealing pressure, sealing pressure ramp
time, maximum forming pressure, forming pressure ramp time,
coining pressure, and coining pressure ramp time. The pressure
increased from its current value to the set value over the ramp time
by following an S-curve. The S-curve was created by fitting a sine
wave over the acceleration and deceleration portions of a constant
acceleration trapezoidal curve, as shown in Fig. 8. Once the pressure curve ramped to the desired pressure, a constant pressure was
maintained until the next stage of the forming process was reached.
The ram velocity–time history displayed in Fig. 8 was generated based on six parameters: sealing acceleration distance, sealing
maximum velocity, sealing travel, forming acceleration distance,
forming maximum velocity, and forming travel. The ram velocity increased from zero to the specified velocity over the specified
Fig. 9. Snapshots of the von Mises stress history are shown as a function of time. The inset defines the pressure and velocity for each snapshot in time.
J. Crapps et al. / Journal of Materials Processing Technology 210 (2010) 1726–1737
1733
Fig. 10. The effective plastic strain is shown in the color contour for different simulations with the forming pressure decreasing for each simulation. The inset
pressure–time history plots illustrate the different simulations with six different pressure histories. Clearly, the higher the gradient between 2 and 4 s yields greater plastic
strains.
acceleration distance by following an S-curve similar to the pressure curve. The ram S-curve was also a sine wave fit over a constant
acceleration trapezoidal curve. Once the ram velocity curve accelerated to the specified velocity, a constant velocity was maintained
until the deceleration portion of the curve was reached. The total
ram travel distance was equal to the area under the ram velocity curve. The control system (physical and simulated) calculated
the area under the velocity–time history curve. If this area was
less than the total travel distance, a constant velocity was added
to the ram velocity–time history to reach the specified travel distance. If this area was more than the total travel distance, the
acceleration distance was reassigned to half the total travel distance and the maximum velocity was reassigned to the velocity
which would have been reached using the original acceleration
distance.
Now that the ram movements have been discussed, assumptions related to the other components will be explained. The die
was fixed in such a manner that no displacements or rotations were
admitted in any direction. The bucking plate was force controlled
as it was designed to inhibit the branch of the tee so fracture would
not occur. The bucking plate was able to move as close as 1.524 mm
from the outer diameter of the copper billet. Master-slave surface
contacts existed between the copper blank and the die, ram, and
bucking plate.
3.4. Preliminary simulation results
Fig. 9 shows snapshots at different times of the von Mises stress
in the deforming copper blank during the hydroforming simulation.
These results were obtained using EMMI without damage (ϕ0 = 0).
These snapshots depict the connection between the pressure- and
velocity–time histories with the stress contours and deforming
geometry of the tube throughout the forming process. Clearly, the
developed finite element model and corresponding solution procedure with four different steps (see Fig. 7) made the numerical
simulation very robust, capturing well the details of the hydroforming process.
To better understand the effect of the white water pressure on
the formation of the tee branch, a number of simulations were run
where the transition time from the sealing pressure to the forming
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J. Crapps et al. / Journal of Materials Processing Technology 210 (2010) 1726–1737
Fig. 11. The damage level is shown in the color contour for different simulations with the forming pressure decreasing for each simulation. The inset pressure–time
history plots illustrate the different simulations with six different pressure histories. Clearly, the higher pressure gradient between 2 and 4 s yields greater damage. Note
the apparent volume increase of the elements with high levels of damage (porosity). High values of damage strongly affect the deformation patterns of the hydroformed
tube.
pressure (pressure gradient) was varied from approximately 2 to
6 s. Again, this study was performed with EMMI without damage.
The results from these simulations are presented in Fig. 10, where
contour plots of the effective plastic strains at the end of the process are given. Two observations are important from this figure.
First, extremely large strains occur when the pressure gradient is
high. Simulations 1–3 all show effective strains up to 500%. Second, wrinkling (instability from localized buckling) starts to occur
as the forming pressure is decreased. Simulation 6 shows the wrinkling starting in the radius of the tee. On the other hand, simulation
4 and 5 indicates that the pressure gradients used for these cases
are adequate to form the tee branch, although simulation 5 uses
a lower forming pressure. This numerical study illustrates the fact
that there exists a window of optimum processing parameters that
produces well-processed tee-tubes.
Another group of simulations with identical boundary conditions as those described above was performed using the EMMI
model with damage. The initial value of damage was set to
ϕ0 = 0.001. The results from these simulations, given in Fig. 11,
are presented in terms of contour plot of damage (void volume
fraction) at the end of the forming process. As shown by this figure, the pressure gradient has a strong effect on the evolution
of damage, with the model predicting porosity levels of up to
one where the pressure transients are short (simulations 1–4).
Note that this may indicate bursting or fracture at the top of the
copper tee. Also note that damage levels that induced fracture
J. Crapps et al. / Journal of Materials Processing Technology 210 (2010) 1726–1737
1735
Fig. 13. Comparison of finite element simulation results with the experimental
results from a processed hydroformed tee-tube that induced wrinkling.
Fig. 12. Comparison of the finite element simulation results with the experimental
results from a well-processed hydroformed tee-tube.
align with the extremely large strains exhibited in Fig. 10. It is
important to point out here that, as the hydroforming process is
a pressure-driven process, one may expect that the tube blank will
experience high stress triaxialities (ratio of the hydrostatic stress
to von Mises equivalent stress) in zones with high tensile hydrostatic stresses. In fact, the simulations showed maximum stress
triaxiality values ranging from 1.3 to 6, depending on the boundary conditions. Since the void growth rule used by EMMI depends
exponentially on this quantity, high void growth rates are expected
to happen for hydroforming boundary conditions inducing large
tensile stresses. This behavior has been observed in the numerical
simulations.
The above simulations evaluated the robustness of the finite element model constructed as well as the general capability of the
EMMI model to predict the large plastic deformations and ductile
damage evolution during the hydroforming process. As validation
experiments for the damage aspects of the hydrofoming process
is lacking (effect of processing parameters on the failure/fracture
of copper tee-tubes), the next section of the paper mainly
focuses on validating both the process model and the plasticity
aspects of EMMI with specific experimental data collected during
the study.
4. Simulation/validation results and discussion
Once the aforementioned hydroforming simulations were performed, an experimental validation study was conducted at a
hydroforming plant. One check was to compare the material
flow patterns of points in the copper blank with that obtained
from the numerical simulations. In the plant, the forming process was set by five main parameters: ram forming travel, forming
Table 3
Validation experiments for hydroforming simulation.
Test
Ram travel (mm)
Pressure (MPa)
Form P ramp (s)
Buck position (mm)
Buck force (N)a
Nominal
Buck force (stunt)
38.862
38.862
38.862
38.862
38.862
34.474
34.474
34.474
34.474
34.474
5.5
5.5
5.5
5.5
5.5
10.16
10.16
10.16
10.16
10.16
1001
1112
1223
1334
278
Buck force (blow)
a
Force values are for a quarter hydroforming model.
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J. Crapps et al. / Journal of Materials Processing Technology 210 (2010) 1726–1737
Fig. 14. Comparison of material point profiles for the hydroforming validation
experiments and the associated numerical simulation results with a varying friction coefficient. Note that when the coefficient of friction was = 0.075 the results
compared extremely well.
Fig. 15. Comparison of material point profiles for the hydroforming validation
experiments and the associated numerical simulation results with a bucking resistance. Note that all cases compare very well.
5. Conclusions
pressure, forming pressure ramp time, bucking punch position,
and bucking punch force. For the validation study, some of
these parameters were varied in the physical process as presented by Table 3. Fig. 12 shows clearly that the numerical
predictions compared well with the experimental results for a
well-processed tee-tube. Fig. 13 also shows a great comparison
of the simulation results with the experiment for a wrinkled
tee-tube.
For the simulations related to each set of conditions presented
in Table 3, the physical profile of the tubes were measured and
compared to the simulation profiles. A simulation of the nominal case was used to estimate the friction coefficient for the
hydroforming process using a methodology similar to Ngaile et
al. (2004). Fig. 14 presents the physical and simulation profiles of
the overall tee and the bottom section of the tee for the nominal
case. The comparison shows that the friction coefficient providing
the best behavior had a value of = 0.075. This value corroborates results found in Ngaile et al. (2004). The small difference
in the bottom wall thickness is probably due to alignment and
tolerance issues in the hydroforming tooling. Using the coefficient of friction of 0.075, Fig. 15 shows that the overall and
bottom profile for each case included in the validation experiments agrees well with those obtained from the hydroforming
experiments.
A modeling-experimental study was performed on a complex
hydroforming process using a non-dimensionalized internal state
variable history model (EMMI) that captures strain rate and multiaxial stress state effects. The model constants were determined
from uniaxial stress–strain responses and the predictive model was
used to evaluate different processing histories. The model was validated from experimental results obtained from in-plant processes.
The excellent comparisons indicate that the model can be used to
optimize the process space to eliminate wrinkling or fracture. The
developed model is used in a second article to perform a parametric
study of the copper T-shape tube hydroforming process.
Acknowledgement
This work was performed under the auspices of the Center for
Advanced Vehicular Systems (CAVS) at Missisissippi State University.
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