INTRODUCTION
Sheet metal forming is one of the most widely used manufacturing processes for the fabrication of a wide range of products in many industries. The reason behind sheet metal forming gaining a lot of attention in modern technology is due to the ease with which metal may be formed into useful shapes by plastic deformation processes, in which the volume and mass of the metal are conserved and metal is displaced from one location to another (Kishor and Kumar, 2002). Deep drawing is one of the extensively used sheet metal forming processes in the industries to have mass production of cup shaped components in a very short time. In deep drawing, a flat blank of sheet metal is shaped by the action of a punch forcing the metal into a die cavity. Sheet metal can be formed using simple procedures, such as bending, or they can be very complex, such as deep drawing of nonaxisymmetrical shapes. Sheet metal or sheet metal products can also be divided into two or more parts using blanking and piercing. Sheet metal is usually coldformed, but in certain cases, such as bending or deep drawing, material can also be heated, usually only locally, in order to increase its formability. Since, sheet metal is formed using tensile or tensilepressure forming, tools used are less loaded than during bulk forming. Product accuracy, especially for thick sheet metal, is not great, since surfaces are partially free and material is formed along the easiest natural path.
Springback and side wall curl: In sheet metal forming processes, at
the end of the operation the blank being formed conforms closely to the shape
of tools. After the load is released and the tools are removed ushaped undergoes
significant changes of its geometrical and shape parameters (Bogdan, 2003).
This phenomenon is called a springback. Springback parameters are mainly influenced
by the following factors:
• 
Material type 
• 
Punch and die radii 
• 
Initial clearance 
• 
Friction conditions 
• 
Blankholder force 
• 
Sheet thickness 
• 
Punch speed rate 
• 
Constitutive behavior in plastic zone 

During the forming process of the ushaped part, the sidewall suffers complicated
bending and stretching phenomena, the stress distribution on the side near the
die is subjected to tensile stress and the side near the punch is subjected
to compressing stress, which would promote a residual bending moment a result
in sidewall curl. Introducing a considerable blankholder force into the forming
process is useful in removing sidewall curl. When the holdown force is increased,
namely increasing the flow resistance of the material, the stress distribution
through the thickness of the sidewall may be turned to tensile stress over the
whole section. Accordingly, springback directions of both sides become consistent,
which is conducive to decreasing shape distortion (Samuel, 2000).
MATERIALS AND METHODS
Theoretical modeleling: Springback in the stretch bending of elastic
nonlinear research hardening material is consider an initially straight beam
of a rectangular cross section of width b and depth h, which is loaded by pure
bending moment M and a tensile axial force N as shown in Fig.
1.
The material of the beams is considered to be elastic exponential research
hardening, i.e. obeying the stressstrain relationship and is represented mathematically
by:
where, σ, σ_{y} are the stress and yield stress, respectively,
e and e_{y} are the engineering strain and engineering yield strain,
respectively, e_{ult} is the ultimate tensile engineering strain, n
is the research hardening index. E and E_{p }are the elastic and plastic
modulii, respectively. Assuming that the plane sections of the beam remain plane
during the elastoplastic stretch bending i.e., the strain at a point in the
beam is proportional to its distance from the neutral axis and the total deflection
is small compared to the length of the beam, so that the additional bending
moment caused by N can be neglected. For a different combination of M and N,
the stress distribution across the section of a rectangular beam is derived
for the following cases.
• 
Totally elastic case, for which no in the beam is strained
beyond the yield point (Fig. 2a) 
• 
Primary plastic case, for which layer on one side of the neutral axis
are beyond the yield condition (Fig. 2b) 
• 
Secondary plastic case, for which layer on both sides of neutral axis
are strained beyond yielding (Fig. 2c) 


Fig. 1: 
An initially straight beam of rectangular cross section, loaded
by pure bending moment and an axial force 

These three cases are illustrated schematically in Fig. 2.
It is convenient to define the following initial yield quantities, bending moment
Me, axial force Ne and curvature κe of the section as:
Evaluation of N and M in plain strain deep drawing;
Holddown force N_{h
}Friction force F_{h} = μN_{h
}N = F_{h}
For the deep drawing with zero radius of die; it will be assumed that the radius
of bend is equal to half thickness plus the clearance of die.
R = h/2+clearance
E = is the Young modulus of sheet metal
then,
M = (Ebh3/12)/ (h/2+clearance)
The dimensionless moment m, stretching force nr and curvature φ are defined
as:
The nondimensional quantities for stress geometry are γ = c/(h/2), δ
= d/(h/2) and f = (1δ+γ) where, c is the distance between the neutral
axis (N.A) and the yielding layer and d is the distance between N.A and centeroidal
axis C.A, which are shown in Fig. 2.
This is the case, when the strain at the outer layer is greater than e_{y}
and the strain at the inner layer is smaller or equals to e_{y}. Referring
to Fig. 2b and Eq. 2.

Fig. 2: 
The stress and strain distribution across a section of beam
of elastic exponential research hardening material. a): Wholly elastic regime
(E); b): Primary plastic regime (P_{I}) and c): Secondary plastic
regimes (P_{II} and P`_{II}) 

we get,
where, μ is the ratio of the plastic modulus Ep to the elastic modulus.
The strain is linearly proportional to the distance from N.A. and the stress
at the outer layer (σ_{o}) when, z = h/2, Fig. 2b,
is therefore,
The stress distribution for the thickness is shown in Fig. 2b.
When, δ = 0 which is the special case where pure bending is applied, Eq.
3 is reduced to
Which is the same expression obtained by Johnson and Yu (1981) for pure bending
only of elastic exponential work hardening material. In the secondary plastic
regimes, PII and P`II, this is the case where the strain in the inner and outer
layers is greater than ey as
well as the strain in the outer layer. Referring to Fig. 2c
and the strain at z = c, ey = c/R = σy/E, i.e. 1/R = σy/ (Exc). At
z = h/2 + d:
and
at z = h/2 +d, the stress σi and strain ei at thinner layer can be written
as:
e_{i} = c/R = (h/2+d)/E = σy
(1+δ)/γ 


For the case, when the N.A inside the cross section (Fig. 2c).
The moment equilibrium equation through the section of beam is:
and the axial load equilibrium equation for a section of the beam is:
Then, after integrating and arrangement that found:
The dimensionless curvature φ of this regime is simplified to
φ = 1/γ = m/μ
φ^{F} = φφ^{E}

(8) 

where, φ^{F} = κ^{F}/κ_{e }is the nondimensional
final curvature of the beam, φ is the elasoplastic curvature determined
by Eq. 8, φ^{E} is the elastic curvature.
Since,
after found the value of the nondimensional final curvature of the beam φ^{F}
we can find the value of κ_{F} from Eq. 2 and
the value of arc from Eq. 9. Since,
then calculate the angle value θ by substitute in Eq. 11:
Experimental research: The major objective of the experimental research
is to design a die and a punch to obtain the ushaped specimen then determine
the springback ratio and side wall curl of sheet metal specimens. Then selection
of the materials to be used for create the specimens of the metal sheet is an
important initial step in the design process. Have been chosen for create of
the sheet of galvanized steel, brass and aluminum. The specimens must be fit
the die and punch with a suitable clearance. It should be a rectangular sheet
of 20 mm of width and 150 mm of length with 0.51 mm thickness.

Fig. 3: 
Springback device with details, a): Springback instrument,
b): Device components 

Springback device design: Springback testing device shown in Fig.
3a was fully designed and manufactured locally. The main parameters were
design with new ideas to overcome the difficulties of Springback testing and
these main parameters as listed:
Die and punch design: The die and the punch are made of steel to be rigged enough in order to have a negligible deflection, the dimension selected to be fit into the hydraulic compression device, which already exists. The Punch and die clearance was adjusted for sheets of different thickness to maintain a clearance to thickness ratio of (12) (Jang and Thomson, 1989). This value was chosen as a standard condition because, it provided a gap large enough not to cause reverse bending of punch and die clearance on springback and sidewall curl.
Hold down force technique: The compression forces were applied by means
of the hydraulic compression apparatus, which acts on the punch, the screw works
as a converter, it converts the torque into force with some loses because of
the friction between the contacts areas, to overcome this problem, Eq.
12 was used.
where, F is the half of the hold down force, K is the spring constant and x
is the displacement of the screw.
Device components: The device as shown in Fig. 3b
is consisted of the following components.
Compression device: A compression device type PHYWE, model D3300 where, used to compress the punch into the die with axial force with selecting the speed rate.
Die: The frame made from steel (405) channel 100x60x60 mm, which is very rigid.
Screw: Four screws (M6) are used to control hold down force on the specimen.
Covers: Two clamps made of steel to transfer the hold down force from the springs to the specimen with a specific coefficient of friction.
Punch: The punch also made from steel (405) with dimension 26x60x30 mm.
Spring: four springs to convert the nut displacement to a force with an easy way of calculations.
Stead: the steed used is (M 20), which is used to fixed the punch with the hydraulic press.
Pressure gauge: A pressure gauge that read the applied axial force on the punch.
Dial gauge: the dial gauge, which is used to read the axial deflection of the punch by the mean of the compressed fluid in the compression device.
Digital camera: A PC digital camera was used to record the readings of the load gauges and the dial gauge by creating a movie for them throughout the test, by pausing this movie; it is possible to draw the forcedeflection curve.
Effective parameters
Punch position: The punch position is an important parameter in this
study, which is effect directly on the shape of the specimen, so, in this study
a central position was selected to make a symmetrical shape.
Punch speed rate: The hydraulic press the punch with different speed of punching (0.34 and 0.48 mm sec^{1}).
Rolling direction: Sheet metals that exhibit different flow strengths in different directions in the plain of the sheet are defined as having planar anisotropy. Parallel, perpendicular and 45° to the rolling direction represent the three vectors of the planar anisotropy. Anisotropy has a great effect on the bending limit with the relative differences in yield strength.
Friction conditions: This prevents the tension across the face of the punch from increasing sufficiently to stretch the material over the face of the punch. The effect of friction is to reduce the tension at the nose and spread the strain.
Testing method and procedure:
Applying load: After the specimen fixed on the instrument where,
holddown force is acting on the sheet by fixed the screw into the spring and
then acting it on the covers with the requirement displacement, Hence, the peripheral
parts of the specimen are kept in place as shown in Fig. 4a.

Fig. 4: 
Applying and releasing load, a): Loading specimen, b): Releasing
hold down force 

• 
Fixed the die and punch in the compression device. The punch
is moved downwards the specimen till it reaches the sheet 
• 
The punch is now in contact with the sheet and the sheet is drawn through
the opening in the die. It slides over the die edge. As the punch proceeds
downwards the outer radius of the research piece is reduced. In this process,
the specimen is formed through stretching in the drawing 
• 
At the end of processes the blank being formed conforms closely to the
shape of die 

Releasing the load: After the load is released and the tools are removed,
the ushaped part undergoes significant changes of its geometrical and shape
parameters.
To unfasten the ushaped specimen out of the device the following procedures
are followed:
• 
Returning the compression device to its initial position after
the final end of the punch stroke by open the screw of reset tool 
• 
Freeing the specimen from the hold down force by removing one screw from
each side of the die perpendicular to the site of view of the specimen 
• 
Rotating the two covers with 180° away of the punch as shown in (Fig.
4b) 
• 
Finally, the formed component (ushape specimen) will be released directly
from the die 

RESULTS AND DISCUSSION
The theoretical and experimental tests, which were described earlier are listed and discussed in details now. The validity of the theoretical modeling results has been checked by comparing it with the experimental test results.
Experimental modeling results: The results of a Springback tests are presented here. The data was obtained by hand plotting technique, using the milimetric pad after the specimen shape was plotted on the study and then the data was measured by using protractor and vernier.
Radius of curvature estimation: The interpolating polynomial was used
to find the equation of side wall curl curve that show in Fig.
4 that choosing m point out of given (n+1) point we could pass a^{n}
(m1) degree polynomial through these m point. It follows that a^{n}
(m1) degree interpolation gives us a^{n} (m1) degree interpolation
polynomial (Louis, 2006).

Fig. 5: 
Parameters to quantify Springback, a): With rolling direction,
b): Normal rolling direction 

Then, for sidewall curve substitute (m = 6) in Eq. 13:
MATLAB program interpolating polynomial was written to find the coefficients
of the polynomial by substituting the coordinate of different six points that
are taken from the side wall curl arc. Differentiated Eq. 14
and substitute in Eq. 15 (Meriam and Kriage, 2007) to obtain
an approximate value to the side wall radius of curvature.
where, ρ_{xy} is the radius of curvature at any point.
Die and punch angels estimation: The die angle and punch angle were measured from the ushaped of specimen as shown in Fig. 5, by draw the tangential line of sidewall arc and crossed it with the tangential line of the part of a specimen formed from the holdown force. These springback parameters whose variation was observed during experimental research are as:
θ_{1} the die angle between the bottom of die and the sidewall. θ_{2} the punch angle between the effect of holdown force and the sidewall.
The effective factors are:
• 
Holdown force values (200, 600, 800 and 1100) (N) 
• 
Thickness 0.5 and 1.0 mm of brass, aluminum and galvanized steel specimens 
• 
Speed rate of punch force 0.34 and 0.48 mm sec–1 
• 
With and normal rolling direction 
• 
Lubricant with oil SAE 50 


Fig. 6: 
Brass (0.5 mm) without lubricant, a): With rolling direction,
b): Normal rolling direction 

And a complete set of the test results for all types of specimens under these
effective factors are tabulated in Table 1, the graphical
relationships were shown in Fig. 6 and 7
and the comparing results as shown in Fig. 8 and 9,
which show good correlations in relationship between the experimental and theoretical
results.
The neutral axis moves towards the external side of the specimen, which is
towards the die. This is due to the higher forces required to bend the thicker
specimens. The increase is due to the increasing friction that is developed
between the specimen and the die and the punch. Due to the increased friction
forces the amount of sliding between the specimen and the die is reduced and
higher deformations occur for the thicker specimens. This increases the flat
length required.

Fig. 7: 
Normal direction of rolling in speed (0.34 mm sec^{1})
without lubricant a): Thickness 0.5 mm, b): Thickness 1.0 mm 

The variation of springback results is as follows:
• 
The die angle θ_{1} 
• 
Has decreased by 20% with the increase of the holdown force 
• 
Has increased by 26.66% with the increase lubricant oil 

Table 1: 
Springback results of 0.5 mm Brass without lubricant 



Fig. 8: 
Comparing the results of 0.5 mm galvanized under effect of
HDF 


Fig. 9: 
Comparing the results of 1.0 mm galvanized under effect of
HDF 


• 
Has decreased by 15.38% with the change in rolling direction
from with to normal 

• 
Has decreased by 7.14% with the increase punch speed rate 


• 
Has decreased by 21.07% with the increase of the holdown force 

• 
Has increased by 22.72% with the increase lubricant oil 

• 
Has decreased by 10% with the change in rolling direction from with to
normal 

• 
Has decreased by 2.3% with the increase punch speed rate 

• 
The sidewall curvature ρ 


• 
Has decreased by 11% with the increase of the holdown force 

• 
Has increased by 81.8% with the increase lubricant oil 

• 
Has decreased by 7% with the change in rolling direction from with to
normal 

• 
Has decreased by 10% with the increase punch speed rate 

By using MATLAB program, a polynomial function could be estimate the relationship
between all the parameters to find the value of slops and curvature with fixed
dimensions:
Die angle = 1.8519 x3  2.9630 x2  1.0926 x + 13.3222
Punch angle = 2.7778 x3 + 8.6111 x2 10.4444 x + 21.7667
Radius of curvature = 1.1111 x3  1.7778 x2+ 1.3444 x + 5.0933



where, x is the holdown force value in (N), to obtain the accurate results
that substituted any values of holdown force between the ranges (2001100 N)
with multiply the equations by the percentage values of any effects parameters
on the results.
CONCLUSION
The following conclusions were drawn from the study of all factors that govern
the evolution of the stress state in sheet material have a direct influence
on the amount of Springback. The remaining conclusions are derived from overall
shape change:
• 
Due to the extent of the affected area the shape change associated
with simple bending, springback is much smaller than that related to sidewall
curl 
• 
Applying tension drastically reduces springback, although, a drop region
of smaller dependence appears for back forces, i.e., the die angle and punch
angle has decrease with the increase of speed rate of punch force 
• 
Lubricant has increase Springback and side wall curl in sheet metal forming 
• 
A rolling direction has a little effect on the springback values that
decrease when change rolling direction from 090° 
• 
Die angle and punch angle has decrease with increase of holdown force 
• 
The percentage of comparing results between theoretical and experimental
is (1015%) 
• 
When, thickness increases the die and punch angle decrease but radius
of curvature increase 
