mathematical modeling and finite element analysis of superplastic forming of ti 6al 4v alloy in a stepped rectangular die

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Procedia Engineering 64 (2013) 1209 – 1218

..International Conference On DESIGN AND MANUFACTURING, IConDM 2013

Mathematical Modeling and Finite Element Analysis of
Superplastic Forming of Ti-6Al-4V Alloy in a Stepped
Rectangular Die
M.Balasubramanian1*, K.Ramanathan2, V.S.Senthil kumar3
1

Assistant Professor, Department of Mechanical Engineering, Anna University, University College of Engineering -Ramanathapuram Campus,
Ramanathapuram-623513, Tamilnadu, India.
2
Assistant Professor, Department of Mechanical Engineering, A.C. College of Engg &Tech, Karaikudi- 630 004 Tamilnadu, India.
3
Associate Professor, Department of Mechanical Engineering, College of Engineering, Guindy Campus, Anna University, Chennai-600 025,
Tamilnadu, India.

Abstract
Superplastic forming has become a viable process in manufacturing of aircraft and automobile parts such as compressor
blades, window frames and seat structures, turbine disc etc., which require relatively low tooling and assembly cost. In this
paper, the attempt was made to analyze the Ti-6Al-4V alloy sheet using a stepped rectangular die by superplastic blow
forming technique. This alloy is most suitable material for producing complex shapes using superplastic forming methods. The
forming characteristics of thickness distribution, bulge forming time and optimum pressure with and without die entry radius
and friction coefficient in a two step rectangular die have been analyzed by the theoretical model and numerical simulation
using Finite Element Method (Abaqus).
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Key Words: Finite Element method; Mathematical modeling; Stepped Rectangular; Ti-6Al-4V; Superplastic forming process.
___________________________________________________________________________________________________________________

1. Introduction
The superplastic forming is a valuable tool for fabrication of complex parts used in the aircraft and automobile
industries. Superplastic forming of the sheet metal has been used to produce complex shapes and integrated
structure that are often light weight and stronger than the assembled components. Superplasticity is a property of
certain metallic materials, which enable them to achieve very high elongation of 1000% without necking in hot

Corresponding Authors: Tel: +91-04567-291599, fax: +91-04567-291699. Email : *,
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1877-7058 © 2013 The Authors. Published by Elsevier Ltd.
Selection and peer-review under responsibility of the organizing and review committee of IConDM 2013
doi:10.1016/j.proeng.2013.09.200


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M. Balasubramanian et al. / Procedia Engineering 64 (2013) 1209 – 1218

tensile test under certain condition and also this material undergo extreme elongation at the proper temperature and
strain rate. Superplastic deformation is carried out very close to isothermal conditions under controlled strain rate.
The optimum strain rate varies with the superplastic material, which is usually in the range of 0.001 s ¯1 to
0.00001s¯1. This is attributed to the viscous material behaviour exhibited by some metals and alloys with very
fine and stable grain structure at temperature above 0.5 Tm (Tm-melting point of materials). Few materials like
Ti-6Al-4V alloy undergo extensive tensile plastic deformation prior to failure under a specific temperature and
particular strain-rate.
Ghosh and Hamilton (1980) used the plane strain condition to explore the shape of the die on the optimized
pressurization profile during blow forming into a rectangular die. Jovane (1968) used a uniform-deformation
method to analyze the relationship between the optimum pressurization profile and the strain-rate sensitivity during
the blow forming of a circular diaphragm. Hwang et al. (1997) developed a generalized mathematical model
considering uniform and non-uniform thinning in the free bulge region to examine the optimized pressurization
profile and thickness distribution of the product in blow forming into a circular die. Padmanabhan and Davies
(1980) achieved long elongation at slow strain rate at temperature 0.5 T m. Viswanathan et al. (1980) investigated
the theoretical and experimental models of the thermo pressure forming process of the Ti-6Al-4V alloy into a
hemispherical shape. Viswanathan et al. (1990) analysed macro, micro and re-entrant shape in a Ti-6Al-4V alloy
to optimize the forming pressure, time and thickness distribution.
Yogesha and Bhattacharya (2011) studied superplastic deformation capability of the Ti-Al-Mn alloy by

thermoforming route. SenthilKumar et al. (2006) analyzed the finite element modeling of superplastic forming of
AA7475 aluminium alloy in a hemispherical die. Chandra and Chandy (1991) used a finite element analysis
model choosing the membrane element model for the superplastic forming process in a box with a complex shape.
Bonet et al. (1994) developed a finite element analysis model using incremental flow formulation in thick and
thin sheet components. Xing et al. (2004) developed a rigid- viscoplastic finite element program, to predict the
microstructure variation to improve the uniformity of wall thickness. Mimaroglu and Yenihaya (2003) analysed
the superplastic forming process under constant strain rate by the ANSYS finite element analysis code, parametric
design language and ANSYS-visco108 element. Giuliano (2008) considered four-node, isoparametric and
arbitrary quadrilateral elements for Finite element analysis in a Ti-6Al-4V alloy. Chen et al. (2001) used the
continuum element for finite element analysis in Ti-6Al-4V alloy.
Balasubramanian et al. (2004) has developed a theoretical model and C++ coding in a long rectangular die and
analysed superplastic parameters like radius of curvature, bulge forming time, thickness distribution and pressure
profile for 8090 Al-Li alloy. Many work have been carried out in the related field. but only less work has been
reported on Ti-6Al-4V alloy. To best of our knowledge there is no literature focused on two and more than two
stages in a rectangular die under plane strain condition using titanium alloy. Hence in this paper an attempt has
been made for two stages in a long rectangular die with plane strain condition using Ti-6Al-4V alloy.
Superplastic forming process have been done by a simple theoretical model and by numerical analysis using finite
element method (FEM- Abaqus) simulation with accurate prediction of the deformation characteristics.
Nomenclature
Al
D2
h0
k
l1
Mn
p
R
Ri+1
t
Ti

W1

Aluminum
Half the die depth in stage two (mm)
Original sheet thickness (mm)
Material constant (MPa sm)
Length of die in second stage (mm)
Manganese
Forming pressure (MPa)
Radius of curvature (mm)
Decrement in curvature (mm)
Forming time (sec)
Titanium
Half width of the die (mm)

D1
h
hi+1
l1
m
n
pi+1
Ri
S
ti+1
V
W2

Depth of die in first stage (mm)
Current thickness (mm)

Decrement in thickness (mm)
Length of die in first stage (mm)
Strain rate sensitivity index
Strain hardening index
Pressure increment (MPa)
Radius of curvature in first stage (mm)
Arc length (mm)
Increment in time (sec)
Vanadium
Quarter the die width (mm)


M. Balasubramanian et al. / Procedia Engineering 64 (2013) 1209 – 1218

Xi
Xi+1
Yj

Instantaneous in semi width in first stage (mm)

1211

Xj

Instantaneous in semi width in
second stage (mm)
Decrement in semi width length (mm)
Yi
Instantaneous in depth of die in
first stage (mm)

Instantaneous in depth in second stage (mm)
Yi+1
Decrement in depth of die (mm)
Reduction in width of die (mm)
Reduction in depth of die (mm)
Stress (N/mm2)
Stress in width direction (N/mm2)
w
Strain
Strain rate (per sec)
Strain rate in width direction (per sec)
Effective strain rate (per sec)
Angle suspected between radius of curvature and axis line (degree)
Decrement of angle suspected (degree)

2. Theoretical modeling
2.1 Superplastic forming process
Argon Gas

Before Applying
Pressure

(a)

Vent hole

(b)

After Applying
Pressure

(c)

(d)

Fig. 1. (a), (b), (c) & (d) Different stages of pressure blow forming technique in a stepped rectangular die.

Many number of metal forming process such as pressure forming, vacuum forming, thermo forming, deep
drawing, etc have been developed in recent years. Pressure forming is the most widely used method for forming of
superplastic metal into desired components shown in Fig. 1. In superplastic forming process a material is heated to
the superplastic temperature within a closed sealed die, and inert gas pressure was applied, sheet to take the shape
of the pattern. The flow stress of the material during deformation, increases rapidly with increase in pressure.
In order to simulate mathematically, the pressure profile, thickness distribution and forming time in the
superplastic forming process, the numerous constitutive equations have been proposed to characterize the material
flow stress response. The flow stress
for the superplastic material can be expressed as Eq.(1)
(1)


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M. Balasubramanian et al. / Procedia Engineering 64 (2013) 1209 – 1218

2.2 Basic assumptions
The following basic assumptions have been made during the theoretical modeling of the superplastic forming
process, (i) The material is isotropic and incompressible, (ii) The diaphragm is rigidly clamped at the periphery of
the die, (iii) Process is assumed to be plane strain condition, (iv) The specimen thickness is very small when
compared with the die radius, so that bending and shearing effects are negligible.
2.3 Superplastic blow forming process and geometric model
R

R

D1

D1
S

d

S

D1

D2

W1
W2
(b)

W1
(a)

R

y=D2

R
D1

Yi

Yi+1=D2

S

S
D2
W2

Xi

(d)

(c)

R1
Yi+1

W2

S1

Yj

Stage 1
R2

Xj

Stage 2

S2

(e)
Fig. 2. (a), (b), (c) , (d) & (e) Illustration of different stages of blow forming.

D1


M. Balasubramanian et al. / Procedia Engineering 64 (2013) 1209 – 1218

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The Fig. 2 clearly evident that, geometric relationship established to predict the thickness variation, radius of
curvature, arc length, time required to form the curvature and forming pressure during both the step of bulge
forming.
The mathematical relationship [1] is obtained from above geometric blow forming process diagram. In this
theoretical analysis , it is assumed that the depth (D1) of the die is equal to half of the width (W1) of the die in step
one (D1=W1) and step two (D2=W2).

R

(2a)
(2b)
(2c)
(2d)

L

d

(2e)

W1
Fig. 3 Geometric configuration of radius of curvature.

From Fig. 3, the radius of curvature is obtained by the Eq. (2e)
Arc length of bulge is described by the Eq. (3)
(3)
The forming time is calculated in each stage by Eq. (4)
(4)
The current thickness of the sheet during blow forming in each step is obtained by the Eq. (5)
(5)
The sheet is treated as a membrane during forming, the forming pressure is obtained by the Eq. (6)
(6)
Using the above equations, the various superplastic forming parameters are analyzed at every stage of forming
until the profile reaches the bottom of the die.
Subsequently, the forming takes towards the edge direction in both the steps. Assume positive decrement (
)
in width direction and positive decrement ( ) in depth direction during the lengths contacted on the bottom and
sidewall respectively during each stage of processing. Using Y j =Yi+1 Xj = Xi+1
and pressure increment are found from Eqs. (7), (8)& (9).

Ǧ

Ǧ

(7)
(8)
(9)

The time, thickness and pressure computation are carried out in this manner until profile reaches the edge of the
die. Same equation is used to find all parameter in the second step of rectangular shape.


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3. Finite Element Modeling
3.1 FEM model
Superplastic blow forming is a complicated process involving large strain, large deformation and material
nonlinearity. Usually deformation is dependent on boundary conditions. Consequently, the numerical analysis of a
highly nonlinear system presents formidable computational problems. Fortunately, the superplastic behaviours of
materials are characterized by the dependency of the flow stress upon the strain rate, which allows the material to
be described as rigid visco-plastic. Therefore, the simulation of superplastic blow forming can be performed using
the creep strain rate control scheme within FEM (Abaqus). The die and sheet model of quarter stepped rectangular
is shown in Fig. 4.

Fig. 4. FEM model for rectangular sheet.

The finite element simulation in a sheet metal with stepped rectangular geometry, the first step depth
D1 = 14 mm, width 2W1 = 28 mm and length l1 = 120 mm and second step D2 = 7 mm, width 2W2 = 14 mm and
length l2 = 106 mm with 3 mm flange all around it.
The initial dimension of the blank is 126 mm x 34 mm x 1.6 mm; the blank was rigidly clamped on all its
edges. The finite element mesh was generated using brick element in a rectangular sheet. The modified Newton
Raphson method adopted for solving non-linear equation in Abaqus. The material constants of k = 250 MPasm,
T = 927°C and m = 0.58 chosen for Ti-6Al-4V alloy in a numerical simulation analysis.
The nodes of element have three degrees of freedom i.e. X, Y and Z direction, the finite element model and
boundary condition nodes on the blank outer edge had all their degree of freedom constrained. All nodes of the die
surface were totally restricted for any movement in any direction. Pressure has applied to the blank surface in the

Y direction as a distributed load, now several load steps corresponding to each operational procedure are carefully
modeled to obtain an accurate simulation of a superplastic blow forming process in FEM (Abaqus).
3.2 Material selection
Titanium alloys can be used in the fabrication of airframe control surface and small scale structural elements
where low weight and high stiffness are required. Ti-6Al-4V alloy is used for the theoretical modeling and finite
element simulation of the superplastic forming process. Table 1 and Table 2 shows the composition and
mechanical properties of Ti-6Al-4V alloy.


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Table 1. Composition of Ti-6Al-4V Alloy
__________________________________________________________________________________
Component
Ti
Al
V
___________________________________________________________________________________
% of weight
90
06
04
___________________________________________________________________________________
Table 2. Mechanical properties of Ti-6Al-4V alloy
___________________________________________________________________________________
S.No
Mechanical properties
Value

___________________________________________________________________________________
1

Yield Strength

924 Mpa

2

Ultimate Strength

993 Mpa

3

Melting point

1500 to 1600°C

4

Modulus of Elasticity

113.8 GPa

5
Poiss
0.342
____________________________________________________________________________________

3.4 Blow forming components at different stages
The Fig. 5. (a),(b),(c) & (d) Different stages of blow forming of sheet into the stepped rectangular die.
Fig. 5 (a) shows that the full rectangular sheet, before applying boundary and load conditions. Fig. 5 (d) shows
that after completion of blow forming process.

(a)

(b)

(c)

(d)

Fig. 5. (a),(b),(c) & (d) Different stages of blow forming of Ti-6Al-4V sheet into a stepped rectangular die.


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4. Results and discussion
A simple mathematical modeling of superplastic forming of two stepped rectangular box has been developed
and the finite element package is used to predict the superplastic forming parameters such as the thickness
distribution, forming time and optimization of pressure profile.
4.1 Variation of forming pressure as a function of forming time

0.7
0.6
0.5
0.4

0.3
0.2
0.1

FEA(ABAQUS)
Theoretical

0

20
40
60
Forming time (Min)

80

Forming pressure (MPa)

Forming Pressure (MPa)

Superplastic forming depends on the gas pressure and time. The forming pressure with respect to forming time
is shown in Fig. 6.
From Fig. 6, observed that, the rate of change in pressure initially increases then slightly decreases and further
rapidly increases. This observation is due to the rate of change of the thickness which is less than rate of change of
the radius. The forming of the sheet continues, the rate of change of thickness increases while that of the radius
decreases, and pressure reduced to continue the constant flow stress. Once the sheet contacts die surface, the rate of
change of the radius again dominates in both the stages, and a rapid pressure increase obtained.
0.7
0.8
0.6

0.5
0.3
0.2
0.1

0.6
0.5
0.4
0.3
r = 1 mm
m
r = 2 mm
m

0
0

10

20 30 40 50 60
Forming time (Min)

15

30

45

60

75

90

70

Fig. 8. Illustration of effective of optimum pressure as a function of
forming time at different die entry radius. initial thickness of 1.6 mm.

Fig. 7. Illustration of effect of pressure profile as a function
of forming time at different friction coefficient.

Thickness distribution (mm)

Optimum pressure (Mpa)

0.7

0.1

0

Forming time(Min)

Fig. 6. Theoretically predicted forming pressure as a function of forming
time, at initial thickness of 1.6 mm.

0.2

μ=0.0
μ=0.3
μ=0.5

0.4

1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0

r = 1 mm
r = 2 mm
r = 3 mm

0

10

20 30 40 50 60
Forming Time (Min)

70

Fig. 9. Illustration of effective of thickness distribution as a
function of forming time at different die entry radius

4.2 Effect of pressure profile as a function off forming time at different friction coefficient
The friction is widely recognized as an important factor for affecting the thinning of superplastic forming
components. For an initial study, the friction coefficient is assumed uniformly along the contact surface. In
addition, it is reasonable to assume that the friction coefficient ranges from 0.0 to 0.5. The effect of the frictional
coefficient between the die and sheet during superplastic forming is shown in Fig. 7.


M. Balasubramanian et al. / Procedia Engineering 64 (2013) 1209 – 1218

1217

In the free bulged region, the forming pressure initially rises and remains constant at different friction
coefficients. After the bulge profile touches side or bottom wall the friction coefficient has take place.
The Fig .7 shows that the forming pressure decreases while the friction coefficient increases. The forming time
needed to complete the blow forming process and also increases with increasing friction coefficient.
4.3 Effects of optimum pressure as a function of forming time at different die entry radius by FEA
The Fig. 8 shows the optimum pressure as a function of time with respect to different die entry radius. This
profile indicates that, the forming time is decreasing to maintain constant strain-rate deformation with increasing
die entry radius. During the flow formation to maintain constant strain-rate deformation with increasing die entry
radius, the need of forming pressure was slightly decreased. The optimum pressure is obtained when the die entry
radius and radius of the corner increases.
4.4 Effects of thickness distribution as a function of forming time at different die entry radius by FEA
The thickness distribution and forming time were changes with respect to die entry radius as shown in Fig. 9.
This picture clearly conclude that, the thickness distribution is increasing to maintain constant strain-rate
deformation with increasing die entry radius. The forming time rapidly decrease and thickness distribution values
increase when the die entry radius increases. This abrupt thinning is due to the large tension exerted upon the sheet
with free bulged region. As the free bulged region begins to make contact with the wall in both the steps, this rapid

thinning become more profound when the die entry radius increases.
4.5 Effective
f
of thickness distribution from centre along the die profile

Fig. 10. Illustration of thickness distribution along the die surface profile at initial thickness of 1.6 mm.

The Thickness distributions are measured along with die surface and it is represented in Fig. 10. The thinning is
measured from bottom centre point to top flange along die surface. The degree of thinning over the die profile can
be calculated to a good accuracy at different die entry radius. From the Fig. 10, more thinning at second step
bottom corner and first step bottom corner was observed. It indicates that more thinning occurs in the bottom
corner compared to the rest of the part.
5. Conclusion
The mathematical modeling and Finite Element analysis of superplastic forming of Ti-6Al-4V alloy in a
stepped rectangular die leads to the following conclusions.
1.

The pressure increases rapidly when the rate of change of radius is greater than the rate of change of
thickness.


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M. Balasubramanian et al. / Procedia Engineering 64 (2013) 1209 – 1218

2.
3.
4.
5.

The forming pressure requirement was decreased to maintain constant strain-rate deformation with
increasing die entry radius.
At a given forming temperature, thickness distribution varies with increasing die entry radius.
Optimum pressure need decreases with increase in friction coefficient.
More thinning is found at both the bottom corner of the rectangular die.

6. References
[1].

Ghosh. A.K, Hamilton.C.H, 1980. Superplastic forming of a long rectangular box section analysis
and experiment, modeling fundamentals and applications to metals, ASM, Metals Park, OH,
pp 303-329.
[2]. Jovane.F, 1968. An approximate analysis of the superplastic forming of a thin circular diagram theory
and experiments, International Journals of Mechanical Science, vol 10, pp 403-427.
[3]. Hwang.Y.M, Yang.J.S, Chen.T.R, Huang J.C, 1997. Analysis of superplastic sheet metal forming in
circular closed die considering non uniform thinning, Journals of Materials Processing technology,
vol 65, pp 215-227.
[4]. Padmanabhan.K.A and Davies G.J, 1980. Superplastisity, Materials Research and Engineering,
Springer- Verlag, Berlin, Vol 2, pp 1-6.
[5]. Viswanathan.D, Venkatasamy.S, Padmanabhan.K.A, 1980. Theoretical and experimental studies on
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[6]. Viswanathan.D, Venkatasamy.S,
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