INTRODUCTION
Hydroforming Deep Drawing (HDD) is one of the metal forming processes that
is used in industry to produce complex sheets with high Limiting Drawing Ratio
(LDR). Schematic of cylindrical cup drawing with HDD process is shown in Fig.
1. A pressurized fluid is employed in front of the workpiece. As the punch
travels, the workpiece begins to deform into a cylindrical cup (Kandil,
2003).
Some of the advantages of sheet hydroforming are, improving the material formability,
reduction of friction force, the accuracy of the forming part and the reduction
of forming stages because of improvement of Limiting Drawing Ratio (LDR) (SooIk
et al., 2006; Parsa and Darbandi, 2008; Lang
et al., 2005a).
Analysis of tearing phenomenon in hydroforming was studied by Zhang
et al. (2000), Lang et al. (2005b)
and Dachang et al. (2005). Generally, two kinds
of material failure caused by inappropriate fluid pressure were identified.
The failure by wrinkling at the lip area (The area that the blank is in contact
with die and blank holder) results from insufficient fluid pressure and the
failure by rupture on the top of the cup results from excessive fluid pressure
(Sywei Lo et al., 1993).
Numerous researchers have attempted to explain theoretically the critical condition
of rupture in hydroforming processes. Yossifon and Tirosh
(1985), Tirosh and Hazut (1989) and Yossifon
and Tirosh (1991) predicted rupture by using the criterion of plane strain
failure and wrinkling instability by energy method. They also obtained tearing
and wrinkling diagrams for a die with radial pressure. Sywei
Lo et al. (1993) extended the results of Yossifon
and Tirosh (1991) for hemispherical cups. Wu et al.
(2004) and Khandeparkar and Liewald (2008) obtained
rupture and wrinkling diagrams for stepped punches by finite element simulation
and experiments. Thiruvarudchelvan and Tan (2006) performed
theoretical analysis and experimental approach from hydraulic pressureassisted
deep drawing process. Hama et al. (2007) developed
an elastoplastic finite element method for the sheet hydroforming of elliptical
cups.

Fig.1: 
Hydroforming Deep Drawing (HDD) process 
In this study, a suitable punchstroke pressure path was obtained theoretically
that avoids rupture in HDD process. Also, using the finite element simulation,
the limiting pressure path was obtained. The results obtained showed that
the theoretical pressure path is an upper limit of the tearing diagram.
This study was conducted in Babol university of technology in Iran country
in 20072008 years.
BASIC THEORETICAL FORMULATIONS
A number of assumptions are made in this analysis that are:
• 
The thickness of the workpiece remains constant through
out the process 
• 
The principal strain axes do not rotate 
• 
The tresca yield criterion is satisfied and the fluid pressure p is smaller
than the radial stress σ_{r} and the tangential stress σ_{θ}.
This assumption yields σ_{r}σ_{θ} = σ_{e}. 
For axisymmetric problems the polar equilibrium equation in the rim area is
(Tirosh and Hazut, 1989):
where, f (p) is the friction force in the rim area. By using the tresca
criterion and a power law for the material properties, Eq.
1 can be rewritten as:
By using the normal anisotropy of the material in the formulation, the
equivalent strain rate is denoted as:
Since it was assumed that the axis of strain does not rotate and by considering
that the material follows volume constancy in the plastic deformation,
the effective strain ε_{e} is obtained by integrating Eq.
3:
By substituting Eq. 4 into Eq. 2, we
have:
Referring to Fig. 2 and substituting Eq.
5 for tension in area 1 we have:
By simplifying Eq. 6 for σ_{r}^{(1)}
we have:

Fig. 2: 
Cylindrical sheet hydroforming process 
The value of strain in area 1 is obtained by:
Where:
Here,
is heaviside unit function. In the same manner, the stress in area 2
is as follows:
in which the strain in area 2, ε_{r}^{(2)}, is:
Where:
In Eq. 12, β is an angle shown in Fig.
3 and is obtained by:

Fig.3: 
Geometry of β angle in analytical equations 
If the bending stress is neglected and the radial stress σ_{r}
is calculated for the pure radial drawing case, equilibrium of area (2)
become:
ANALYSIS OF TEARING IN HYDROFORMING DEEP DRAWING
Tearing occurred at the upper part of the punch; just at the beginning of draw
workpiece. It is cause by the firm contact between the workpiece and punch due
to circumferential compressive fluid pressure (Yossifon and
Tirosh, 1985). This area is the transition between regions 2 and 3.The rate
of tangential strain at the wall of the punch is zero. It mines:
By combining Eq. 15 and 3 and solving,
then we have:
Yossifon and Tirosh (1985) predicted the instability
of anisotropic material under biaxial plane strain conditions as fallows:
Where:
So, if Eq. 17 is in Eq. 10 will have:

Fig. 4: 
The finite element model for the analysis of HDD in
Abaqus software 
Obviously, necking and rupture occur when the radial load reaches a maximum
value.
By equating Eq. 10 and 19 we have:
By considering Eq. 14 and 21 we can obtain
the fluid pressure causing rupture in the workpiece.
FINITE ELEMENT SIMULATION
The commercial finite element software ABAQUS 6.6 was used for
the simulation. The finite element model created in the software is shown
in Fig. 4.
In the simulations, the tools (punch, pressure chamber components and
blank holder) were considered as rigid, while the sheet was considered
to be deformable material. Coulomb friction equation was used to model
the frictional condition between the blank and the tools. Due to symmetry,
only half of the die and blank crosssection were modeled. In order to
model the liquid, a uniform pressure distribution was used to apply the
fluid pressure directly to the blank on the die opening. To introduce
the pressure into the software use subroutine, called Vd load in ABAQUS,
was used. The punch, die and blank holder were meshed with RAX2 (2node
linear axisymmetric rigid link for use in axisymmetric planar geometries)
and the blank was meshed with SAX1 (2node linear axisymmetric thin or
thick shell).
Table 1: 
Material properties of the finite
element simulation 

Table 2: 
Material processes of the finite
element simulation 

In order to define the rupture diagram, several counter pressurepunch displacement
were prescribed for the simulation. The rupture criterion used in this paper
is the critical effective strain at instability (Yossifon
and Tirosh, 1985), as:
The material properties and the process parameters are given in Table
1 and 2, respectively.
RESULTS AND DISCUSSION
The tearing diagrams obtained by the analytical and finite element
simulation are shown in Fig. 5.
Both of the methods suggested that a counterpressure history with a
somewhat smaller pressurization at the initial stage and a larger one
at the later stage would normally result in a proper cylindrical cup product.
As you see, the finite element simulation predicts limited zone compared
with analytical equation. It means that analytical path is an upper limit
to the tear diagram. Figure 6 investigates the effects
of anisotropy, drawing ratio, sheet thickness and strain hardening component
on tearing diagram.
In HDD the die radius does not always coincide the blank profile radius.
The blank may move with variable contact with the die radius when counterpressure
going up so different values of blank profile radii obtained in HDD process
for different die profile radii. This variation is shown in Fig.
7.
As it is see in Fig. 8, when the die profile radius is less
than 2 mm the blank profile radius coincides the die curvature. But by growing
die profile radii the blank profile radius changes along HDD process. Figure
8 has shows schematically this phenomenon.

Fig. 5: 
Tearing diagram obtained from analytical
approach and finite element simulation 

Fig. 6: 
Effect of, (a) Anisotropy, (b)
Drawing ratio, (c) Sheet thickness and (d) Strain hardening component,
on tearing diagram in HDD process 

Fig. 7: 
Schematic of blank profile radius
die profile radius, (a) R_{d}>2 and (b) R_{d} = 1 

Fig. 8: 
Blank profile radius in HDD process
for different dies curvature 
CONCLUSION
The results of theoretical and finite element simulations of the
HDD process showed that theoretical tearing pressure is always higher
than tearing pressure obtained by finite element simulation.