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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 the deformation 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 cost and time in the sheet metal-forming process. Direct inverse design method such as Ideal Forming (Chang and Richmond, Int J Mech Sci 1992; 34(7) and (8): 575}91 and 617}33) [7, 8] for optimum blank shape could play an important role to give a basic idea to designer at the initial die design stage of the sheet metal-forming process. However, it is di$cult to predict an exact optimum blank without fracture and wrinkling using only the design code because of the insu$cient accuracy. Therefore, the combination of a design code and an analysis code enables the accurate blank design. In this paper, a new blank design method has been suggested as an e!ective tool combining the ideal forming theory with a deformation path iteration method based on FE analysis. The method consists of two stages: the initial blank design stage and the optimization stage of blank design. The "rst stage generated a trial blank from the ideal forming theory. Then, an optimum blank of the target shape is obtained with the aid of the deformation path iteration method which has been newly proposed 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 C X GH x GH r
Cauchy strain tensor nodal coordinates in the initial state nodal coordinates in the "nal state 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
1218
n M K = R K *;G j G e G e , e * d ##
S.H. Park et al. / International Journal of Mechanical Sciences 41 (1999) 1217}1232
strain hardening exponent exponent used in Hill's new yield criterion sti!ness coe$cient plastic work external load vector tangential sti!ness displacement vector principal value of Cauchy strain tensor principal strain rate e!ective strain and e!ective strain rate shape error amounts small value euclidean vector norm
1. Introduction Sheet metal forming may involve stretching, drawing, bending or various combinations of these basic processes. Deformation defects such as shape error, fracture, and wrinkling, etc. have been easily found in sheet metal forming due to large deformations. Besides the process control, an appropriate method of process design is necessary to get a sound product. Generally, a suitable set of process parameters are determined by the trial-and-error method, because the process involves various 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 of investigations have been carried out to get the optimum blank shape which could be deformed into the net shape [1}9]. The direct inverse design method has been reported to have got an initial blank from the "nal shape 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 theory to get an initial blank shape. Since, these studies [6}8] did not consider the real forming conditions such as blank holder forces, friction forces, and tool geometry, etc., the calculated blank shape had some shape errors. Recently, it was reported that the process parameters of real forming process should be appropriately considered for more precise blank shape. Barlat et al. [9] suggested an inverse design approach using a mathematical technique to get a blank taking into consideration the actions of the tools in contact with the sheet. Analytical methods using the FE analysis code were also developed [3}5]. However, a large computation time was required in order to obtain a precise blank shape. In the present work, a combination of the ideal forming theory and the so-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 ideal forming 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 element positions and accordingly the initial blank shape are obtained as solutions. However, the initial blank obtained from the ideal forming theory is not generally the optimum one in the real forming operations, 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
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modi"cation of the blank shape. In order to obtain a more accurate blank, the following optimum blank design stage is introduced on the basis of the iterative FE analysis using the e!ective computational algorithm. In the second stage, a deformed shape is calculated from the initial blank with the help of the rigidplastic FE analysis code [10]. Obviously, the deformed shape computed from the initial blank, as a solution of ideal forming theory, may involve some shape errors when compared with the target shape. 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. In this paper, the initial blank, obtained as a solution of ideal forming theory is the starting point for the next optimum blank design stage. The basic assumptions of the ideal forming theory used in this 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 following equations:
+\ + ,
+ + e "D "eR #eR "+\*D "eR !eR "+\
(1)
where . D " [2(1#r)] +, D "(1#2r)\ +\ The e!ective strain can be obtained by integrating the e!ective strain rate along the minimum work path. This path is achievable only when the principal stretch lines maintain their directions with respect to the material during deformation and the ratio of principal true strain rates is also kept constant [7, 8]. R
eNQ dt
eN "
(2)
Then, the e!ective strain is "nally given as
+ +\
+\ + ,
j + eN "D "ln(j ) j )"+\#D ln j where j and j are the principal values of Cauchy strain, C .
(3)
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The e!ective stress is also obtained by the power law as follows: pN "K(e #eN )L. (4) Then, the internal plastic work, = are calculated by using the e!ective strain and e!ective stress de"ned in Eqs. (3) and (4).
4 pN ) eN d< .
="
(5)
The ideal forming theory involves solving the following equation to get an initial blank. This means that the total plastic work must be optimized in the initial blank state. d= "0 for i"1, 2, (6) dX G where X is any component of the coordinate system in the initial state. G The Newton}Raphson method is employed to solve Eq. (6), and the detailed formulation is shown 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 to design the optimum blank shape. Table 1 Schematic diagram of the deformation path iteration method.
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The necessary and su$cient conditions of the stress "eld for the rigid-plastic sheet metal to be in equilibrium at time t #q is given from the following virtual work principle; i.e., O (7) d=O " pN d(*eN )t dA"d=O .
Using the Newton}Raphson method, it can be expressed as K*;G"R!FG\,
(8)
where K is a tangential sti!ness matrix; R is an external load vector, *;G is a displacement vector, and FG\ means the internal force of the (i!1)th iteration. Eq. (8) is iterated until the following condition is satis"ed: #*;#/#;#)d,
(9)
where # ) # 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, and real deformation path, etc. are not considered at all. Then, the computed initial blank inevitably involved some shape errors when compared with the optimum blank. A deformation path iteration method is proposed to minimize the shape error in the optimum blank 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 analysis code, the redundant area is subtracted from the initial blank by the same amount of volume along the deformation path. On the other hand, some shape errors exist along the boundary line of the target 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 target shape after the whole deformation. If any shape error is still remaining, the whole procedure is repeated until the error is smaller than the given error bound (in this study, 0.5 mm). Table 1 shows a detailed algorithm of the proposed method and the volume addition and subtraction procedures are outlined in Fig. 1. As shown in Fig. 1, an average thickness is considered to calculate the volume of *abc in the deformed shape and the initial thickness of sheet is used for *ABC in the blank. The calculated error volume is subtracted or added based on incompressibility at the blank periphery along the strain 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., the addition 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 the extrapolation method using the Lagrange's formula, as given in Eq. (10). ¸(x)" y # y # y #2# y # y L\ L\ L L where (x!x )(x!x )2(x!x ) L , y "f (x ),2, y "f (x )
" L L G (x !x )(x !x )2(x !x ) G G G L
(10)
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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(* ) is introduced to de"ne the geometrical deviation quantitatively. The geometrical shape error, * is de"ned as root mean square of the shape di!erence between the target shape and the deformed shape as in Eq. (11):
1 , * " d , G N
(11)
where d is the distance between the target shape and the deformed shape along the deformation G paths, and N is the number of nodal points along the boundary of the blank.
S.H. Park et al. / International Journal of Mechanical Sciences 41 (1999) 1217}1232
Fig. 2. Deformation paths obtained by using FE analysis.
Fig. 3. De"nition of shape error, * .
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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 path iteration method, the formability should be checked for the next step, because the formability is a!ected by the blank shape. In this paper, the Keeler's approach [12] is used to examine the formability. The method is very simple but it has provided useful results in the sheet metal forming process. The FLD (see Fig. 4) can be calculated by the following empirical equation. F¸D "(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 strain hardening exponent. The forming limit curve is shifted along the major strain axis based on the point at which the minor strain is zero (FLD ) 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 is shown in Fig. 5, and the dimensions and the properties of the sheet material used in the simulation are given as shown in Table 2. Computation has been carried out for only one-quarter of a square cup due to symmetry considering all the necessary process variables. The computed initial blank based on the ideal forming theory is shown in Fig. 6. The computation time took about 110 s using the HP Workstation-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
S.H. Park et al. / International Journal of Mechanical Sciences 41 (1999) 1217}1232
Fig. 5. Target shape.
Fig. 6. Initial blank shape calculated from the ideal forming theory.
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Fig. 7. Tool shape for FE analysis.
Table 2 Process and material parameters used in the simulation of square cup deep drawing Height Width of cup Width of #ange Corner radius (wall to wall) Corner radius (wall to #ange) Sheet material Stress-strain relation (MPa) Lankford value Thickness of sheet Coulomb coe$cient of friction Blank holding force
25 mm 40 mm;40 mm 10 mm 5 mm 5 mm Mild steel p"565.32 (0.007117#e) r "1.77 0.78 mm 0.1 100 kgf
forming theory. Therefore, computation is also carried out to get an optimum blank using the analysis code. Fig. 7 shows the schematic view of tools for FE analysis. The initial blank obtained from the ideal forming theory and its deformed shape by the analysis code are shown in Fig. 8. As shown in Fig. 8, the geometrical shape error can be found (* "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 deformed contour exists. The amount of geometrical shape error can be fed back for modifying the initial
S.H. Park et al. / International Journal of Mechanical Sciences 41 (1999) 1217}1232
Fig. 8. (a) Initial blank obtained from the ideal forming theory and (b) its deformed shape.
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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 and its deformed shape. In Fig. 9, it is easily seen that the shape error is reduced as compared to the case of the initial blank trial (* "0.696 mm). The geometrical shape error (* ) de"ned in Eq. (12) is still larger than the error bound (0.5 mm). The "rst modi"ed blank should then be corrected again 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
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Fig. 10. (a) 2nd modi"ed blank and (b) its deformed shape.
lies within the error bound. The value of shape error, * is 0.117 mm. Now, the shape error is shown to be reduced within the error bound with two modi"cation steps by using the proposed method. 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
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Fig. 11. (a) The comparison of the deformed shape for various stages. (b) the comparison of the deformed shape for various stages (Top view).
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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-forming problems. In the previous research work [13], it has been shown that a blank shape has a direct in#uence on the formability. In order to compare the formability of the optimum blank with a general square blank having the same area and the thickness of the sheet as the optimum blank, the comparison is shown in the FLD curve as in Fig. 12. In this "gure, the case of optimum blank shows the improved formability as compared with the square blank.
5. Conclusions A new method of optimum blank design has been proposed by using the ideal forming theory and the deformation path iteration method. The method was integrated in the "nite element modeling of sheet metal-forming process. The design procedure is composed of two stages; The "rst stage is the initial blank design stage based on the ideal forming theory and the design modi"cation stage 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 to optimize the initial blank. Deep drawing of a square cup has been treated as an example. It has been found out that with two iterations the deformed contour shape becomes almost coincident with the target shape and
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the optimized blank demonstrates the improved formability as compared with the usual square blank. It has been thus shown that the proposed method is an e!ective tool for optimum blank design with improved formability and can be further applied to optimum blank design of other practical sheet metal-forming problems.
Reference [1] Majlessi SA, Lee D. Development of multistage sheet metal forming analysis method. J Mater Shap Technol 1988;6(1):41}54. [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 Res 1985;25(1):15}32. [4] Chung WJ, Kim YJ, Yang DY. Rigid-plastic "nite element analysis of hydrostatic bulging of elliptic diagrams using Hill'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 "nite element 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 1992;34(7):575}91. [8] Chung K, Richmond O. Ideal forming*II. Sheet forming with optimum deformation. Int J Mech Sci 1992;34(8):617}33. [9] Barlet O, Batoz JL, Guo YQ, Mercier F, Naceur H, Knopf-Lenoir C. Optimum design of blank contour using the inverse approach and a mathematical programming technique. Numisheet'96, 1996;178}85. [10] Yoo DJ, Song IS, Yang DY, Lee JH. Rigid-plastic "nite element analysis of sheet metal forming processes using continuous 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. Proceedings of the 3rd I.C.T.P., 1990;3:1311}8.
Optimum blank design in sheet metal forming by the deformation 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 cost and time in the sheet metal-forming process. Direct inverse design method such as Ideal Forming (Chang and Richmond, Int J Mech Sci 1992; 34(7) and (8): 575}91 and 617}33) [7, 8] for optimum blank shape could play an important role to give a basic idea to designer at the initial die design stage of the sheet metal-forming process. However, it is di$cult to predict an exact optimum blank without fracture and wrinkling using only the design code because of the insu$cient accuracy. Therefore, the combination of a design code and an analysis code enables the accurate blank design. In this paper, a new blank design method has been suggested as an e!ective tool combining the ideal forming theory with a deformation path iteration method based on FE analysis. The method consists of two stages: the initial blank design stage and the optimization stage of blank design. The "rst stage generated a trial blank from the ideal forming theory. Then, an optimum blank of the target shape is obtained with the aid of the deformation path iteration method which has been newly proposed to minimize the shape errors at the optimization stage. In order to verify the proposed method, a square cup example was investigated. 1999 Published by Elsevier Science Ltd. All rights reserved. Key words: Optimal blank design; Deep drawing; Ideal forming theory; Path iteration method
Notations C X GH x GH r
Cauchy strain tensor nodal coordinates in the initial state nodal coordinates in the "nal state Lankford value for normal anisotropy
* Corresponding author. 0020-7403/99/$ - 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
1218
n M K = R K *;G j G e G e , e * d ##
S.H. Park et al. / International Journal of Mechanical Sciences 41 (1999) 1217}1232
strain hardening exponent exponent used in Hill's new yield criterion sti!ness coe$cient plastic work external load vector tangential sti!ness displacement vector principal value of Cauchy strain tensor principal strain rate e!ective strain and e!ective strain rate shape error amounts small value euclidean vector norm
1. Introduction Sheet metal forming may involve stretching, drawing, bending or various combinations of these basic processes. Deformation defects such as shape error, fracture, and wrinkling, etc. have been easily found in sheet metal forming due to large deformations. Besides the process control, an appropriate method of process design is necessary to get a sound product. Generally, a suitable set of process parameters are determined by the trial-and-error method, because the process involves various 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 of investigations have been carried out to get the optimum blank shape which could be deformed into the net shape [1}9]. The direct inverse design method has been reported to have got an initial blank from the "nal shape 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 theory to get an initial blank shape. Since, these studies [6}8] did not consider the real forming conditions such as blank holder forces, friction forces, and tool geometry, etc., the calculated blank shape had some shape errors. Recently, it was reported that the process parameters of real forming process should be appropriately considered for more precise blank shape. Barlat et al. [9] suggested an inverse design approach using a mathematical technique to get a blank taking into consideration the actions of the tools in contact with the sheet. Analytical methods using the FE analysis code were also developed [3}5]. However, a large computation time was required in order to obtain a precise blank shape. In the present work, a combination of the ideal forming theory and the so-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 ideal forming 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 element positions and accordingly the initial blank shape are obtained as solutions. However, the initial blank obtained from the ideal forming theory is not generally the optimum one in the real forming operations, 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
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modi"cation of the blank shape. In order to obtain a more accurate blank, the following optimum blank design stage is introduced on the basis of the iterative FE analysis using the e!ective computational algorithm. In the second stage, a deformed shape is calculated from the initial blank with the help of the rigidplastic FE analysis code [10]. Obviously, the deformed shape computed from the initial blank, as a solution of ideal forming theory, may involve some shape errors when compared with the target shape. 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. In this paper, the initial blank, obtained as a solution of ideal forming theory is the starting point for the next optimum blank design stage. The basic assumptions of the ideal forming theory used in this 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 following equations:
+\ + ,
+ + e "D "eR #eR "+\*D "eR !eR "+\
(1)
where . D " [2(1#r)] +, D "(1#2r)\ +\ The e!ective strain can be obtained by integrating the e!ective strain rate along the minimum work path. This path is achievable only when the principal stretch lines maintain their directions with respect to the material during deformation and the ratio of principal true strain rates is also kept constant [7, 8]. R
eNQ dt
eN "
(2)
Then, the e!ective strain is "nally given as
+ +\
+\ + ,
j + eN "D "ln(j ) j )"+\#D ln j where j and j are the principal values of Cauchy strain, C .
(3)
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The e!ective stress is also obtained by the power law as follows: pN "K(e #eN )L. (4) Then, the internal plastic work, = are calculated by using the e!ective strain and e!ective stress de"ned in Eqs. (3) and (4).
4 pN ) eN d< .
="
(5)
The ideal forming theory involves solving the following equation to get an initial blank. This means that the total plastic work must be optimized in the initial blank state. d= "0 for i"1, 2, (6) dX G where X is any component of the coordinate system in the initial state. G The Newton}Raphson method is employed to solve Eq. (6), and the detailed formulation is shown 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 to design the optimum blank shape. Table 1 Schematic diagram of the deformation path iteration method.
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The necessary and su$cient conditions of the stress "eld for the rigid-plastic sheet metal to be in equilibrium at time t #q is given from the following virtual work principle; i.e., O (7) d=O " pN d(*eN )t dA"d=O .
Using the Newton}Raphson method, it can be expressed as K*;G"R!FG\,
(8)
where K is a tangential sti!ness matrix; R is an external load vector, *;G is a displacement vector, and FG\ means the internal force of the (i!1)th iteration. Eq. (8) is iterated until the following condition is satis"ed: #*;#/#;#)d,
(9)
where # ) # 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, and real deformation path, etc. are not considered at all. Then, the computed initial blank inevitably involved some shape errors when compared with the optimum blank. A deformation path iteration method is proposed to minimize the shape error in the optimum blank 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 analysis code, the redundant area is subtracted from the initial blank by the same amount of volume along the deformation path. On the other hand, some shape errors exist along the boundary line of the target 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 target shape after the whole deformation. If any shape error is still remaining, the whole procedure is repeated until the error is smaller than the given error bound (in this study, 0.5 mm). Table 1 shows a detailed algorithm of the proposed method and the volume addition and subtraction procedures are outlined in Fig. 1. As shown in Fig. 1, an average thickness is considered to calculate the volume of *abc in the deformed shape and the initial thickness of sheet is used for *ABC in the blank. The calculated error volume is subtracted or added based on incompressibility at the blank periphery along the strain 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., the addition 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 the extrapolation method using the Lagrange's formula, as given in Eq. (10). ¸(x)" y # y # y #2# y # y L\ L\ L L where (x!x )(x!x )2(x!x ) L , y "f (x ),2, y "f (x )
" L L G (x !x )(x !x )2(x !x ) G G G L
(10)
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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(* ) is introduced to de"ne the geometrical deviation quantitatively. The geometrical shape error, * is de"ned as root mean square of the shape di!erence between the target shape and the deformed shape as in Eq. (11):
1 , * " d , G N
(11)
where d is the distance between the target shape and the deformed shape along the deformation G paths, and N is the number of nodal points along the boundary of the blank.
S.H. Park et al. / International Journal of Mechanical Sciences 41 (1999) 1217}1232
Fig. 2. Deformation paths obtained by using FE analysis.
Fig. 3. De"nition of shape error, * .
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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 path iteration method, the formability should be checked for the next step, because the formability is a!ected by the blank shape. In this paper, the Keeler's approach [12] is used to examine the formability. The method is very simple but it has provided useful results in the sheet metal forming process. The FLD (see Fig. 4) can be calculated by the following empirical equation. F¸D "(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 strain hardening exponent. The forming limit curve is shifted along the major strain axis based on the point at which the minor strain is zero (FLD ) 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 is shown in Fig. 5, and the dimensions and the properties of the sheet material used in the simulation are given as shown in Table 2. Computation has been carried out for only one-quarter of a square cup due to symmetry considering all the necessary process variables. The computed initial blank based on the ideal forming theory is shown in Fig. 6. The computation time took about 110 s using the HP Workstation-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
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Fig. 5. Target shape.
Fig. 6. Initial blank shape calculated from the ideal forming theory.
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Fig. 7. Tool shape for FE analysis.
Table 2 Process and material parameters used in the simulation of square cup deep drawing Height Width of cup Width of #ange Corner radius (wall to wall) Corner radius (wall to #ange) Sheet material Stress-strain relation (MPa) Lankford value Thickness of sheet Coulomb coe$cient of friction Blank holding force
25 mm 40 mm;40 mm 10 mm 5 mm 5 mm Mild steel p"565.32 (0.007117#e) r "1.77 0.78 mm 0.1 100 kgf
forming theory. Therefore, computation is also carried out to get an optimum blank using the analysis code. Fig. 7 shows the schematic view of tools for FE analysis. The initial blank obtained from the ideal forming theory and its deformed shape by the analysis code are shown in Fig. 8. As shown in Fig. 8, the geometrical shape error can be found (* "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 deformed contour exists. The amount of geometrical shape error can be fed back for modifying the initial
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Fig. 8. (a) Initial blank obtained from the ideal forming theory and (b) its deformed shape.
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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 and its deformed shape. In Fig. 9, it is easily seen that the shape error is reduced as compared to the case of the initial blank trial (* "0.696 mm). The geometrical shape error (* ) de"ned in Eq. (12) is still larger than the error bound (0.5 mm). The "rst modi"ed blank should then be corrected again 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
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Fig. 10. (a) 2nd modi"ed blank and (b) its deformed shape.
lies within the error bound. The value of shape error, * is 0.117 mm. Now, the shape error is shown to be reduced within the error bound with two modi"cation steps by using the proposed method. 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
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Fig. 11. (a) The comparison of the deformed shape for various stages. (b) the comparison of the deformed shape for various stages (Top view).
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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-forming problems. In the previous research work [13], it has been shown that a blank shape has a direct in#uence on the formability. In order to compare the formability of the optimum blank with a general square blank having the same area and the thickness of the sheet as the optimum blank, the comparison is shown in the FLD curve as in Fig. 12. In this "gure, the case of optimum blank shows the improved formability as compared with the square blank.
5. Conclusions A new method of optimum blank design has been proposed by using the ideal forming theory and the deformation path iteration method. The method was integrated in the "nite element modeling of sheet metal-forming process. The design procedure is composed of two stages; The "rst stage is the initial blank design stage based on the ideal forming theory and the design modi"cation stage 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 to optimize the initial blank. Deep drawing of a square cup has been treated as an example. It has been found out that with two iterations the deformed contour shape becomes almost coincident with the target shape and
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the optimized blank demonstrates the improved formability as compared with the usual square blank. It has been thus shown that the proposed method is an e!ective tool for optimum blank design with improved formability and can be further applied to optimum blank design of other practical sheet metal-forming problems.
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