Indian J.Sci.Res.1(2) : 410-415, 2014

ISSN:2250-0138(Online)

ISSN : 0976-2876 (Print)

__________________________________ 1Corresponding author

SPRINGBACK REDUCTION IN EXPLOSIVE DEEP DRAWING PROCESS OF A TRAPEZOIDAL CUP

ALI JABBARI1a

aAsistant Professor, Department of Mechanical Engineering, Faculty of Engineering, Arak University, Arak 38156-8-8849, Iran

ABSTRACT

In this paper, a die shape optimization method for reduction of spring back defect due to elastic recovery behavior in explosive deep drawing

process is presented by using the reduced basis technique coupled by finite element and design of experiments methods. The primary

objective of the proposed method is to reduce the enormous number of design variables required to define the initial die shape. The reduced

basis technique is a weighted combination of several basis shapes. The aim of the method is to find the best combination using the weights

for each shape as the design variables. The experimental design of Taguchi method is used to build the approximation model and to perform

optimization. This method is demonstrated on the die shape optimization of deep drawing of a trapezoidal cup.

KEYWORDS: Explosive Deep Drawing Process, Springback, Reduced Basis Technique, Design Of Experiment Method.

Explosive deep drawing process has been used to form

trapezoidal cups from a trapezoidal die shape because of high

deformation rate compared to other production methods,

including casting and machining. However, the earring defect

caused by anisotropy behavior of sheet metal seriously affects

the method performance. Since the earring defect is produced

by the sheet metal anisotropy, it is greatly affected by the

initial die shape. Spring back angle minimization can be

performed using die shape optimization to obtain better stress

distribution and also to reduce the number of production

stages.

The main challenge of current optimization methods is the

number of design variables required for die shape

optimization.

An integrated algorithm is presented in this research and it is

conducted to die shape optimization in a sheet metal Explosive

deep drawing process of a trapezoidal cup. An innovative,

comprehensive way of using an efficient design variables

linking method, termed as reduced basis technique (Jabbari et

al., 2009). is demonstrated for die shape optimization. In the

reduced basis technique, many initial die shapes, called basis

shapes, are combined linearly by assigning weight factors.

Different resultant shapes can be generated by changing their

weight factors. Therefore, the number of design variables

required to define the die shape is reduced to the number of

basis shapes. So, the weights assigned for each basis shapes

are the design variables and the optimization goal is to find the

best possible combination of these weights to minimize spring

back angle.

The algorithm presented in this paper focuses on the Taguchi

design of experiments method which is the combination of

mathematical and statistical techniques used in the empirical

study of relationships and optimization, in which several

independent variables influence a dependent variable or

response.

SPRINGBACK PHENOMENON

This phenomenon is due to the elastic deformation part of

recovery and it affects the final geometry of the output

product. However the analysis of the behavior of matter and

spring back is complicated. Due to the large deformation, high

strain rate, mechanical properties and size of the work piece

interaction interface between the interface and the behavior of

gas bubbles resulting from the explosion of non-linear

dynamics, this issue is very complex. Factors that may be

involved in the return spring blast are: wave transmitter,

increasing thickness, increasing or decreasing the radius of the

punch and matrix spans the width and the behavior of

anisotropic plate. Spring back is an intrinsic property of

deformation which with a proper design, it can be brought to a

minimum.

Figure 1. Spring Back defect.

FINITE ELEMENT ANALYSIS

In this research, sheet metal anisotropy effect in Explosive

deep drawing process has been studied using finite element

ALI JABBARI : SPRINGBACK REDUCTION IN EXPLOSIVE DEEP DRAWING PROCESS OF A TRAPEZOIDAL CUP

Indian J.Sci.Res.1(2) : 410-415, 2014 411

package of ABAQUS. In process modeling, punch, matrix and

die holder considered asrigid parts. Material properties and

size of the die and punch is listed in Table 1. Dynamic/Explicit

Analysis which is appropriate for metal forming problems has

been selected. Shell elements of S4R are used to mesh the die.

The “Partition” command is used to mesh the sheet more

accurate. A view of meshed die is shown in Fig. 2.

Figure 2. A view of meshed die.

Then, the results of Explosive deep drawing process

simulation analyzed. Fig. 3 shows final trapezoidal cup after

Explosive deep drawing process.

Figure 3. Trapezoidal cup due to die deformation.

DIE SHAPE OPTIMIZATION

The generated earrings due to sheet anisotropy is clearly seen

in Figure 3.In order to reduce the height of the sheet shape

earrings, Taguchi design of experiments method is used to

optimize initial die shape. Using Taguchi method to optimize

one or several variables must be examined at several levels,

and a well defined objective function, which is a function of

the height of the earring. The weight ratio of 3 to a maximum

radius ( ), the average radius ( ) minimum radius ( )

were allocated, And the height of the earring in each of the 3

radius, a factor attributed to each of the planes. Table 2 shows

the factors considered. The radius of maximum coefficient is

very low because no amount of uncut sheets and sheets in the

initial state. In other rays of the Rings height, weight

coefficients were considered.

OPTIMIZATION METHOD

Although the reduced basis technique is widely used in metal

forming process, and shape optimization in permanent die

motors (Paul Degarmo., 2003). but it is suggested for die

shape optimization in this work. However, it should be

adopted for shape optimization of Explosive deep drawing

process.

Appropriate starting basis shapes are required to employ the

algorithm to find optimumblank shape design. The problem

can be solved in multiple levels as shown in Fig. 3 in which

the optimization procedure guides the designer progressively

in selecting viable basis shapes. In first Level, the basis shapes

may not be anywhere near to what they are supposed to be, but

by the first set of basis shapes the one can determine a best

combination from the first trial shapes.

Figure 3. Algorithem of design optimization.

CASE STUDY

The initial die shape optimization of trapezoidal cup is

demonstrated in this work. The finite element package

ABAQUS is used to simulate the process and to calculate the

spring back angle in order to conduct DOE. A view of the

trapezoidal cup deep drawing model is shown in Figure 4.

Specifications of the investigated sheet metal and the finial

cup are presented in Table 1.

ALI JABBARI : SPRINGBACK REDUCTION IN EXPLOSIVE DEEP DRAWING PROCESS OF A TRAPEZOIDAL CUP

Indian J.Sci.Res.1(2) : 410-415, 2014 412

Figure 4. A view of Explosive deep drawing process

modeling.

Table 1. Specification of the sheet metal and the final cup.

A B N M MELTING

TEMP

TRANSITION

TEMP

MASS

DENSITY

YOUNG

MODULUS

JOHNSON

COOK

1750 3800.001 0.5 0.32 1811 298 7.89 2000000

D1 D2 D3 D4 D5 MELTING

TEMPERTUR

TRANSITIN

TEMPERTUR

REFERENCE

STRAIN

RATE

JOHNSON

COOK

DAMAGE

-2.2 5.43 -0.47 0.016 0.63 1811 298 1

DETONATI

ON WAVE

SPEED

A B OMEGA R1 R2 MASS

DENSITY

TNT 693000 3737700 37471 0.35 4.15 0.9 1.63

C0 S GAMMA0 MASS

DENSITY

MOHIT 149000 1.79 1.65 1

Figure 5. The selected basis shapes.

2-D FEA simulations of the basis shapes are performed in

ABAQUS software to find the spring back angle for

preliminary analysis as shown in Fig. 5. The spring back

angles of the Basis shapes are 3.81, 2.52, and 0.12 (angel),

respectively. From this preliminary analysis, it can be said that

the Basis 2 is more successful than the other two shapes in

reducing the spring back angle. Therefore, the contribution of

Basis 2 must be more than the other basis shapes, which must

be recognized by the optimizer. Each Basis shape is defined

by one shape variables. These shape variables form the

respective Basis vector. These basis vectors is combined with

the weighting factors, a1, a2and a3 that correspond to each

basis vector based on the following equation

where, and n is number of basis shapes.

ALI JABBARI : SPRINGBACK REDUCTION IN EXPLOSIVE DEEP DRAWING PROCESS OF A TRAPEZOIDAL CUP

Indian J.Sci.Res.1(2) : 410-415, 2014 - 413 -

The reduced basis technique is applied to three basis vectors

and the number of design variables is decreased to three,

which are the weights for each basis vector. By changing these

weights, it is possible to obtain various resultant die shapes for

the optimizer to find the best combinations of these weights.9

DOE points are generated to conduct simulation. All of the

resultant die shapes are scaled to maintain in a limited area.

A DOE/Taguchi approach is used to study the effects of

multiple variables simultaneously. Three factors including

Basis shapes weighting factors will be investigated and their

optimum values will be specified through ANOVA. Based on

known variation of spring back angle with respect to different

factors, each factor is considered to have three levels.

Therefore, an L-9 orthogonal array has been selected to run

the experiments. Table IV shows the factors and their levels

and the layout for the selected array is also presented in Table

V.

Table 2. Weighting factors and their selected value.

FACTOR LEVEL 1 LEVEL 2 LEVEL 3

A1(R=45) 0.2 0.1 0.03

A2(R=46.5) 0.5 0.7 0.9

A3(R=51) 0.3 0.2 0.07

Table 3. A standard L-9 (9 Experiments Runs) array for

three three levels factors.

1 2 3

TRIAL1 1 1 1

TRIAL2 1 2 2

TRIAL3 1 3 3

TRIAL4 2 1 2

TRIAL5 2 2 3

TRIAL6 2 3 1

TRIAL7 3 1 3

TRIAL8 3 2 1

TRIAL9 3 3 2

Angles obtained from experiments described in Table 3 in

Table 4 is given.

Table 4. Angles obtained from experiments.

TRIAL NO. RADIUS

1 47.55

2 47.04

3 46.51

4 47.43

5 46.69

6 47.42

7 46.95

8 47.76

9 48.07

The experiments are carried out using FEA and the spring

back angle obtained from each experiment is listed in Table 5.

Table 5. The values of spring back angle in all 9 trials.

TRIAL NO. SPRING BACK ANGLE (

1 0.44

2 1.62

3 2.52

4 1.23

5 2.31

6 1.24

7 2

8 0.01

9 0.9

Considering spring back angle as cost function, the results are

investigated. The main effects table, which presents the mean

value of spring back angle for each factor at all levels, is

shown in Table 6. ANOVA table for spring back angle is

listed in Table 7.

ALI JABBARI : SPRINGBACK REDUCTION IN EXPLOSIVE DEEP DRAWING PROCESS OF A TRAPEZOIDAL CUP

Indian J.Sci.Res.1(2) : 410-415, 2014 - 414 -

Table 6. The mean value of spring back angle for each

factor at all levels.

FACTOR LEVEL

1

LEVEL

2

LEVEL

3

A1 1.526 1.593 0.969

A2 1.223 1.313 1.553

A3 0.563 1.25 2.276

Table 7. ANOVA table for spring back angle.

FACTOR F S V F PS P(%)

A1 2 0.702 0.351 2.354 0.404 7.173

A2 2 0.174 0.087 0.584 0 0

A3 2 4.461 2.23 14.946 4.162 73.844

OTHER/ERROR 2 0.297 0.148 18.983

TOTAL 9 5.636 100.00%

Predicted optimum combination of factors in this step is

shown in Table 8.

Table 8. Optimum condition for spring back angle.

FACTORS VALUES LEVEL CONTRIBUTION

A1 0.03 3 -0.394

A2 0.5 1 -0.14

A3 0.3 1 -0.8

TOTAL CONTRIBUTION

OF FACTORS

-1.335

GRAND AVERAGE OF

PERFORMANCE

1.363

EXPECTED RESULT AT

OPTIMUM

0.029

As shown in Table 8, it is observed from the analysis of results

that Basis 3 doesn't have considerable effect on the spring

back angle. Since its respective weighting factor in optimum

condition is close to zero. Therefore, the optimum shape of

this process used as a Basis shape for second level.

The optimum values for weighting factor a1, a2 and a3 are

0.05, 0.80, and 0.30, respectively. The spring back angle of the

resultant shape is . It is clear that most of the

contribution is from Basis 2. A comparison of maximum Von

Mises stress for basis shapes and opimal shape is shown in

Fig. 6.

(A=45) (B=46.5) (C=51) (D=47.78)

Figure 6. A comparison of maximum Von Mises stress for

basis shapes and optimal shape.

As shown in Fig. 7, the maximum value of the spring back

angle for Basis shapes are 3.81, 2.52, and 0.12 (degree), which

it has been reduced significantly by this method to .

(B)

(A)

Figure 7. A comparison of initial (A) and

final (B) cup.

.

CONCLUSION

A die shape optimization method for Explosive deep drawing

process is introduced in this paper using the reduced basis

technique. The concept of design optimization process is

introduced, which aids the designer in the selection of

practical basis shapes that will give spring back angle

ALI JABBARI : SPRINGBACK REDUCTION IN EXPLOSIVE DEEP DRAWING PROCESS OF A TRAPEZOIDAL CUP

Indian J.Sci.Res.1(2) : 410-415, 2014 - 415 -

reduction. It is important to mention that if expert knowledge

is available, then practical basis shapes can be selected and the

optimum die shape can be obtained in a single level.

Increasing the number of basis shapes also enables the

designer to obtain a better die shape, but the computation time

also increases to build an approximation model. The reduced

basis method aids in the use of the Taguchi design of

experiments models for optimization. Most die shapes

obtained by this method are practical. The presented algorithm

has been applied on die shape optimization of a trapezoidal

cup as a case study. An optimum die shape has been achieved

by the implemented algorithms, starting from three basis

shapes such as three circular arcs. The spring back angle has

been reduced significantly by this optimization method.

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Shape Optimization of Permanent Die Brushless DC Motors

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Shape Optimization of IPM Motors Using an Integrated

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Jabbari A., Shakeri M., Nabavi Niaki A.; 2010. Iron Pole

Shape Optimization of Permanent Die SynchronousMotors

Using the Reduced Basis Technique, Advances in Iranian

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