Optimum blank design in sheet metal forming by the deformation path iteration method

International Journal of Mechanical Sciences 41 (1999) 1217}1232

Optimum blank design in sheet metal forming by thedeformation path iteration method

S.H. Park!, J.W. Yoon!, D.Y. Yang",*, Y.H. Kim#

! LG Production Engineering Research Center, LG Electronics Inc., Pyungtaek, South Korea" Department of Mechanical Engineering, KAIST, 373-1 Kusong-dong, Yusong-gu, Taejon 305-701, South Korea

# Department of Mechanical Design Engineering, Chungnam Nat+l. Univ., Taejon, South Korea

Received 28 August 1996; received in revised form 7 July 1998

Abstract

Optimum blank design methods have been introduced by many researchers to reduce development costand time in the sheet metal-forming process. Direct inverse design method such as Ideal Forming (Chang andRichmond, Int J Mech Sci 1992; 34(7) and (8): 575}91 and 617}33) [7, 8] for optimum blank shape could playan important role to give a basic idea to designer at the initial die design stage of the sheet metal-formingprocess. However, it is di$cult to predict an exact optimum blank without fracture and wrinkling using onlythe design code because of the insu$cient accuracy. Therefore, the combination of a design code and ananalysis code enables the accurate blank design. In this paper, a new blank design method has been suggestedas an e!ective tool combining the ideal forming theory with a deformation path iteration method based onFE analysis. The method consists of two stages: the initial blank design stage and the optimization stage ofblank design. The "rst stage generated a trial blank from the ideal forming theory. Then, an optimum blankof the target shape is obtained with the aid of the deformation path iteration method which has been newlyproposed to minimize the shape errors at the optimization stage. In order to verify the proposed method,a square cup example was investigated. ( 1999 Published by Elsevier Science Ltd. All rights reserved.

Key words: Optimal blank design; Deep drawing; Ideal forming theory; Path iteration method

Notations

C3 Cauchy strain tensorX

ijnodal coordinates in the initial state

xij

nodal coordinates in the "nal stater Lankford value for normal anisotropy

*Corresponding author.

0020-7403/99/$ - see front matter ( 1999 Published by Elsevier Science Ltd. All rights reserved.PII: S 0 0 2 0 - 7 4 0 3 ( 9 8 ) 0 0 0 8 4 - 8

n strain hardening exponentM exponent used in Hill's new yield criterionK sti!ness coe$cient= plastic workR external load vectorK tangential sti!ness*;i displacement vectorji

principal value of Cauchy strain tensore5i

principal strain ratee6 , e60 e!ective strain and e!ective strain rate*%3303

shape error amountsd small valueE E euclidean vector norm

1. Introduction

Sheet metal forming may involve stretching, drawing, bending or various combinations of thesebasic processes. Deformation defects such as shape error, fracture, and wrinkling, etc. have beeneasily found in sheet metal forming due to large deformations. Besides the process control, anappropriate method of process design is necessary to get a sound product. Generally, a suitable setof process parameters are determined by the trial-and-error method, because the process involvesvarious factors such as material properties, blank shape, friction, geometric shape of a die, etc.Among these factors, the initial blank shape is one of the most important factors and plenty ofinvestigations have been carried out to get the optimum blank shape which could be deformed intothe net shape [1}9].

The direct inverse design method has been reported to have got an initial blank from the "nalshape directly within a moderate computation time. Especially Chung and Richmond [7, 8]proposed a direct design method and its theoretical basis, which is called the ideal forming theoryto get an initial blank shape. Since, these studies [6}8] did not consider the real forming conditionssuch as blank holder forces, friction forces, and tool geometry, etc., the calculated blank shape hadsome shape errors. Recently, it was reported that the process parameters of real forming processshould be appropriately considered for more precise blank shape. Barlat et al. [9] suggested aninverse design approach using a mathematical technique to get a blank taking into considerationthe actions of the tools in contact with the sheet. Analytical methods using the FE analysis codewere also developed [3}5]. However, a large computation time was required in order to obtaina precise blank shape. In the present work, a combination of the ideal forming theory and theso-called deformation path iteration method is proposed.

At the "rst stage, the ideal forming theory is used in order to get an approximate blank. The idealforming design theory requires material elements to deform along the minimum plastic work paths,assuming that such paths provide optimum formability [7, 8]. Then, the initial material elementpositions and accordingly the initial blank shape are obtained as solutions. However, the initialblank obtained from the ideal forming theory is not generally the optimum one in the real formingoperations, because the real conditions are often quite di!erent from the assumed ideal conditions.Then, the initial blank obtained from ideal forming was used as an initial trial geometry for further

1218 S.H. Park et al. / International Journal of Mechanical Sciences 41 (1999) 1217}1232

modi"cation of the blank shape. In order to obtain a more accurate blank, the following optimumblank design stage is introduced on the basis of the iterative FE analysis using the e!ectivecomputational algorithm.

In the second stage, a deformed shape is calculated from the initial blank with the help of the rigid-plastic FE analysis code [10]. Obviously, the deformed shape computed from the initial blank, as asolution of ideal forming theory, may involve some shape errors when compared with the targetshape. In this paper, a deformation path iteration method minimizing the shape error is proposed.A square cup example is employed to show the accuracy and e!ectiveness of the proposed scheme.

2. Initial blank design stage

The ideal forming theory [7, 8] is employed to calculate an initial trial blank for FE analysis. Inthis paper, the initial blank, obtained as a solution of ideal forming theory is the starting point forthe next optimum blank design stage. The basic assumptions of the ideal forming theory used inthis paper are given as follows:1. The deformation path based on minimum plastic work is employed without process parameters

such as friction, blank holding force and lubrication, etc.2. Material is modeled as rigid-plastic material, and normal anisotropy of Hill's new criterion [11]

is employed.

For the case of Hill's new yield theory, the e!ective strain rate can be expressed as the followingequations:

e60"D1CDeR 1#eR

2D

MM~1*D

2DeR1!eR

2D

MM~1D

M~1M

, (1)

where

D1"1

2[2(1#r)]

1M, D

2"(1#2r)~ 1

M~1 .

The e!ective strain can be obtained by integrating the e!ective strain rate along the minimumwork path. This path is achievable only when the principal stretch lines maintain their directionswith respect to the material during deformation and the ratio of principal true strain rates is alsokept constant [7, 8].

eN"Pt

0

eNQ dt (2)

Then, the e!ective strain is "nally given as

eN"D1

D ln(j1) j

2)D

MM~1#D

2 K lnAj1

j2BK

MM~1

M~1M

, (3)

where j1

and j2

are the principal values of Cauchy strain, C3 .

S.H. Park et al. / International Journal of Mechanical Sciences 41 (1999) 1217}1232 1219

The e!ective stress is also obtained by the power law as follows:

pN "K(e0#eN )n. (4)

Then, the internal plastic work,= are calculated by using the e!ective strain and e!ective stressde"ned in Eqs. (3) and (4).

="PV0

pN ) eN d<0. (5)

The ideal forming theory involves solving the following equation to get an initial blank. Thismeans that the total plastic work must be optimized in the initial blank state.

d=dX

i

"0 for i"1, 2, (6)

where Xiis any component of the coordinate system in the initial state.

The Newton}Raphson method is employed to solve Eq. (6), and the detailed formulation isshown in Refs. [7, 8]

3. Optimum blank design stage

In this paper, the rigid-plastic FE-analysis code [10] incorporating the bending e!ect is used todesign the optimum blank shape.

Table 1Schematic diagram of the deformation path iteration method.


The necessary and su$cient conditions of the stress "eld for the rigid-plastic sheet metal to be inequilibrium at time t

0#q is given from the following virtual work principle; i.e.,

d=q*/5"P

qpN d(*eN )t

0dA"d=q

%95. (7)

Using the Newton}Raphson method, it can be expressed as

K*;i"R!Fi~1, (8)

where K is a tangential sti!ness matrix; R is an external load vector, *;i is a displacement vector,and Fi~1 means the internal force of the (i!1)th iteration. Eq. (8) is iterated until the followingcondition is satis"ed:

E*;E/E;E)d , (9)

where E ) E is the Euclidean vector norm, and d is a small-value constant.

3.1. Deformation path iteration method

In the initial blank design stage, the process parameters such as friction, blank holding force, andreal deformation path, etc. are not considered at all. Then, the computed initial blank inevitablyinvolved some shape errors when compared with the optimum blank.

A deformation path iteration method is proposed to minimize the shape error in the optimumblank design stage. If some shape error exists deviating from the boundary line of the target shape,where the target shape is the deformed shape computed from the initial blank using the FE analysiscode, the redundant area is subtracted from the initial blank by the same amount of volume alongthe deformation path. On the other hand, some shape errors exist along the boundary line of thetarget shape, and then the insu$cient volumes are added to the initial blank by the same amount.The modi"ed initial blank is deformed using the FE analysis code again to compare with the targetshape after the whole deformation. If any shape error is still remaining, the whole procedure isrepeated until the error is smaller than the given error bound (in this study, 0.5 mm). Table 1 shows adetailed algorithm of the proposed method and the volume addition and subtraction procedures areoutlined in Fig. 1. As shown in Fig. 1, an average thickness is considered to calculate the volume of *abcin the deformed shape and the initial thickness of sheet is used for *ABC in the blank. The calculatederror volume is subtracted or added based on incompressibility at the blank periphery along thestrain paths obtained from the analysis code. This iterative procedure is convergent in its nature.

In Fig. 2, the deformation paths are known by the analysis code. In the case of Fig. 1a, i.e., theaddition of area by the amount of shape error in the initial blank along the deformation path,the deformation path is not clearly de"ned. In such a case, the unknown paths are obtained by theextrapolation method using the Lagrange's formula, as given in Eq. (10).

¸(x)"/1y1#/

2y2#/

3y3#2#/

n~1yn~1

#/nyn

(10)

where

/i"

(x!x1) (x!x

2)2(x!x

n)

(xi!x

1) (x

i!x

2)2(x

i!x

n), y

1"f (x

1),2, y

n"f (x

n)


Fig. 1. Schematic diagram of shape error volume (a) addition and (b) subtraction.

3.2. Dexnition of shape error

As shown in Fig. 3, a geometrical shape error(*%3303

) is introduced to de"ne the geometricaldeviation quantitatively. The geometrical shape error, *

%3303is de"ned as root mean square of the

shape di!erence between the target shape and the deformed shape as in Eq. (11):

*%3303

"S1N

N+ d2

i, (11)

where diis the distance between the target shape and the deformed shape along the deformation

paths, and N is the number of nodal points along the boundary of the blank.


Fig. 2. Deformation paths obtained by using FE analysis.

Fig. 3. De"nition of shape error, *%3303

.


Fig. 4. Schematic diagram of forming limit curve.

3.3. Forming limit diagram

If the optimum blank is obtained by using the ideal forming theory and the deformation pathiteration method, the formability should be checked for the next step, because the formability isa!ected by the blank shape.

In this paper, the Keeler's approach [12] is used to examine the formability. The method is verysimple but it has provided useful results in the sheet metal forming process. The FLD

0(see Fig. 4)

can be calculated by the following empirical equation.

F¸D0"(23.30#359.0t)]q, (12)

if n*0.21 then q"1, and if n(0.21 then q"n/0.21 where t is the thickness (in) and n the strainhardening exponent. The forming limit curve is shifted along the major strain axis based on thepoint at which the minor strain is zero (FLD

0) as shown in Fig. 4. The major and minor strains

calculated from the FE analysis are utilized to show the formability of the blank.

4. Numerical results and discussion

A square cup is considered to verify the deformation path iteration method. The square cup isshown in Fig. 5, and the dimensions and the properties of the sheet material used in the simulationare given as shown in Table 2.

Computation has been carried out for only one-quarter of a square cup due to symmetryconsidering all the necessary process variables. The computed initial blank based on the idealforming theory is shown in Fig. 6. The computation time took about 110 s using the HPWorkstation-730 system. As described previously, the computed initial blank has signi"cant error,because the process parameters and the real deformation path are not considered in the ideal


Fig. 5. Target shape.

Fig. 6. Initial blank shape calculated from the ideal forming theory.


Fig. 7. Tool shape for FE analysis.

Table 2Process and material parameters used in the simulation of square cup deepdrawing

Height 25 mmWidth of cup 40 mm]40 mmWidth of #ange 10 mmCorner radius (wall to wall) 5 mmCorner radius (wall to #ange) 5 mmSheet material Mild steelStress-strain relation (MPa) p6 "565.32 (0.007117#e)0.2589Lankford value r

!7'"1.77

Thickness of sheet 0.78 mmCoulomb coe$cient of friction 0.1Blank holding force 100 kgf

forming theory. Therefore, computation is also carried out to get an optimum blank using theanalysis code. Fig. 7 shows the schematic view of tools for FE analysis. The initial blank obtainedfrom the ideal forming theory and its deformed shape by the analysis code are shown in Fig. 8. Asshown in Fig. 8, the geometrical shape error can be found (*

%3303"3.669 mm). Due to the use of the

ideal forming theory, the shape error or the di!erence between the target contour and the deformedcontour exists. The amount of geometrical shape error can be fed back for modifying the initial


Fig. 8. (a) Initial blank obtained from the ideal forming theory and (b) its deformed shape.


Fig. 9. (a) 1st modi"ed blank and (b) its deformed shape.

blank design by the deformation path iteration method. Fig. 9 shows the "rst modi"ed blank andits deformed shape. In Fig. 9, it is easily seen that the shape error is reduced as compared to the caseof the initial blank trial (*

%3303"0.696 mm). The geometrical shape error (*

%3303) de"ned in Eq. (12)

is still larger than the error bound (0.5 mm). The "rst modi"ed blank should then be correctedagain by using the same method.

The second modi"ed blank and its deformed shape are shown in Fig. 10. In this case,the deformed shape is almost coincident with the target shape, and the amount of shape error


Fig. 10. (a) 2nd modi"ed blank and (b) its deformed shape.

lies within the error bound. The value of shape error, *%3303

is 0.117 mm. Now, the shape error isshown to be reduced within the error bound with two modi"cation steps by using the proposedmethod.

The value of the geometric shape error vanishes when the number of modi"cation is increased.The deformed shape converges to the target shape with increasing number of modi"cations.Fig. 11 shows the change of the deformed shape for increasing number of modi"cations.From the physical nature of the proposed method, this iterative procedure is intrinsically


Fig. 11. (a) The comparison of the deformed shape for various stages. (b) the comparison of the deformed shape forvarious stages (Top view).


Fig. 12. The comparison of forming limits; from the rectangular blank and the optimum blank.

convergent and the method can be successfully applied for other examples of sheet metal-formingproblems.

In the previous research work [13], it has been shown that a blank shape has a direct in#uenceon the formability. In order to compare the formability of the optimum blank with a general squareblank having the same area and the thickness of the sheet as the optimum blank, the comparison isshown in the FLD curve as in Fig. 12. In this "gure, the case of optimum blank shows the improvedformability as compared with the square blank.

5. Conclusions

A new method of optimum blank design has been proposed by using the ideal forming theoryand the deformation path iteration method. The method was integrated in the "nite elementmodeling of sheet metal-forming process. The design procedure is composed of two stages; The "rststage is the initial blank design stage based on the ideal forming theory and the design modi"cationstage introducing the proposed deformation path iteration method combined with the rigid-plastic"nite element method for sheet metal-forming. The second stage involves the iterative procedure tooptimize the initial blank.

Deep drawing of a square cup has been treated as an example. It has been found out that withtwo iterations the deformed contour shape becomes almost coincident with the target shape and


the optimized blank demonstrates the improved formability as compared with the usual squareblank. It has been thus shown that the proposed method is an e!ective tool for optimum blankdesign with improved formability and can be further applied to optimum blank design of otherpractical sheet metal-forming problems.

Reference

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[2] Batoz JL, Guo YQ, Duroux P, Detraux JM. An e$cient algorithm to estimate the large strains in deep drawing.NUMIFORM'89, 1989;383}8.

[3] Toh CH, Kobayashi S. Deformation analysis and blank design in square cup drawing. Int J Mach Tool Des Res1985;25(1):15}32.

[4] Chung WJ, Kim YJ, Yang DY. Rigid-plastic "nite element analysis of hydrostatic bulging of elliptic diagrams usingHill's new yield criterion. Int J Mech Sci 1989;31:193.

[5] Iseki H, Murota T. On the determination of the optimum blank shape of nonaxisymmetric drawn cup by the "niteelement method. Bull JSME 1986;29(249):1033}40.

[6] Sowerby R, Duncan JL, Chu E. The modelling of sheet metal stampings. Int J Mech Sci 1986;28(7):415}30.[7] Chung K, Richmond O. Ideal forming*I. Homogeneous deformation with minimum plastic work. Int J Mech Sci

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[10] Yoo DJ, Song IS, Yang DY, Lee JH. Rigid-plastic "nite element analysis of sheet metal forming processes usingcontinuous contact treatment and membrane elements incorporating bending e!ect. Int J Mech Sci 1994;36:513.

[11] Hill R. Theoretical plasticity of textured aggregates. Math Proc Camb Phil Soc 1979;85:179}91.[12] Keeler SP. Press shop applications of forming limit diagrams. IDDRG Working Group III, Ann Arbor. MI, 17

October 1976.[13] Ohwue T, Takita M. Analysis of material draw-in from #ange corner during deep drawing of polygnal shells.

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Optimum blank design in sheet metal forming by the deformation path iteration method