Materials and Design 54 (2014) 121–129

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Materials and Design

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Enhancement of dimple formability in sheet metals by 2-step forming

0261-3069/$ - see front matter � 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.matdes.2013.07.085

⇑ Corresponding author. Tel.: +82 2 705 8636; fax: +82 2 712 0799.E-mail address: hylee@sogang.ar.kr (H. Lee).

Minsoo Kim, Sungsik Bang, Hyungyil Lee ⇑, Naksoo Kim, Dongchoul KimDepartment of Mechanical Engineering, Sogang University, Seoul, Republic of Korea

a r t i c l e i n f o a b s t r a c t

Article history:Received 8 April 2013Accepted 26 July 2013Available online 15 August 2013

Keywords:Forming process designMulti-stampingSheet metal stampingFinite element analysisResponse surface method

In this study, a 2-step stamping model with an additional 1st stamping tool is proposed to reduce stamp-ing flaws in the curved parts of dimples in nuclear fuel spacer grids. First, the strains in the curved part ofthe dimple are analyzed and compared with strain solutions for pure bending. A reference 2D FE (finiteelement) model of the 1-step stamping is established, and the corresponding maximum strain isobtained. FE solutions are obtained for various process variable values for the 1st stamping tool usedin the 2-step stamping model. Based on these solutions and applying the RSM (response surface method),strains are expressed as a function of process variables. This function then serves to evaluate optimumprocess variable values. Finally, by transferring these optimum values to a 3D FE model, we confirmthe enhanced formability of the proposed 2-step stamping model.

� 2013 Elsevier Ltd. All rights reserved.

1. Introduction

Support grids for nuclear fuel spacer grids used in light waterreactors serve to maintain integrity of rods in the reactor corethroughout the nuclear fuel’s lifetime at regular core operatingconditions, and guide the cooling flow between the fuel rods toboost discharge of heat from the fuel rods through the coolant[1]. Nuclear fuel spacer grids are manufactured by laser-weldingtogether at right angles thin sheets featuring springs and dimples(Figs. 1(a) and 2). Spring and dimple are formed during the stamp-ing process and protrude from the main body. When the fuel rod isclamped in the nuclear spacer grid, its position is maintained dueto friction between the rod and the stiff dimple and, on the oppo-site side, between the rod and the compressed spring (Fig. 1(b)).[?schematic figure].

During 1-step stamping, deformations are concentrated at thecurved dimple part due to sudden shape changes which may resultin cracking (Fig. 2). To improve formability in stamping or deep-drawing, research has been conducted to develop multi-stepstamping processes. Kim et al. [3,4] analyzed a multi-step proce-dure for rectangular sheets with high aspect ratios. First, draw-backs of existing multi-step stamping tools were identified, andbased on this a new tool with lower susceptibility to crackingand better quality was suggested. Kim et al. [5] analyzed multi-step deep-drawing processes using molybdenum sheets, whichhave a very good high-temperature performance but low formabil-ity. Values were optimized by studying through FEA (finite elementanalysis) the influence of design variables associated with the die

and the punch [on the formability]. Further, Abe et al. [6] and Kuet al. [7] compared final sheet thicknesses obtained by FEA andexperiments, studied the influence of varying the values of designvariables for die and punch and finally assessed the reliability ofthe optimized multi-step procedure by FEA. Huang et al. [8,9] ap-plied FEA and RSM for optimizing the multi-step stamping process.The formability was evaluated by comparing strains and thick-nesses in an FLD (forming limit diagram) [10–13]. Naceur et al.[14] and Zeng et al. [15] have optimized the shape of forming toolsusing response surface analysis to improve formability in deepdrawing and roll forming. They observed the influence of die androll radii on spring-back and confirmed using FEA the reliabilityof the optimized process. Kim et al. [3,4] introduced a deep-draw-ing process for sheets with thicknesses of less than 0.5 mm andsuggested a multi-step procedure to enhance the formability forshapes with high aspect ratio. In this process, the punch and die’saspect ratios are increasing with ongoing process. In this study, a2-step stamping procedure for forming dimples for a spacer gridis suggested that features reduced susceptibility to cracking. The2-step stamping procedure is established by adding to the existing1-step process a preparatory 1st stamping tool of similar finalshape and aspect ratio. Due to the low stamping depth, the sug-gested procedure is cost-effective and shows high productivity.FE solutions are obtained for various process variable values forthe 1st stamping tool used in the 2-step stamping model. Basedon these solutions and applying the response surface method(RSM), strains are expressed as a function of process variables. Thisfunction then serves to evaluate optimum process variable values.These values are then transferred to a 3D FE model. The formabilityof the 2-step procedure is examined [and assessed] based on thestrains (averaged through the thickness) in the weak cross-section

(a)

dimple

spring

(b)

Fig. 1. Schematic of (a) spacer grid set and (b) spacer grid.

Fig. 2. Examples of dimple crack.

Fig. 3. Model for the analysis of pure bending.

Table 1Mechanical properties of Zircaloy-4.

E (GPa) ro (MPa) r

108 364 2.265

122 M. Kim et al. / Materials and Design 54 (2014) 121–129

and the thickness variation at the end and in the middle part of thespecimen. In addition, stresses are compared to the path-indepen-dent FLSD (forming limit stress diagram), and the damage locus isevaluated on grounds of the GTN (Gurson–Tvergaard–Needlman)model.

2. Strains in curved part under pure bending

The FE model for analysis of specimens under pure bending loadis shown in Fig. 3. The specimen’s radius at the inner side is de-noted ri, the radius at the outer side ro and the thickness t. Stressesin circumferential direction (1-direction) are denoted by rh, whilestresses in radial direction (2-direction) are denoted by rr. The ra-dius of curvature of the neutral plane, which undergoes no changein length, is denoted rn and the yield condition at r is [16,17]

rh � rr ¼ þC rn 6 r 6 roð Þ ð1Þ

rh � rr ¼ �C ri 6 r 6 rnð Þ ð2Þ

The equilibrium equation in radial direction for arbitrary radiiof curvature r is therefore

drr

dr¼ rh � rr

r¼ �C

rð3Þ

Applying the boundary condition stating rr = 0 at r = ro andr = ri, and integrating above equation, we get for rr and rh

rr ¼ þC lnrro

; rh ¼ þC 1þ lnrro

� �rn 6 r 6 roð Þ ð4Þ

rr ¼ �C lnrri

; rh ¼ �C 1þ lnrri

� �ri 6 r 6 rnð Þ ð5Þ

Since rr is continuous at r = rn

þC lnrn

ro¼ �C ln

rn

rið6Þ

Finally, the sought-for radius of curvature of the neutral planecan be calculated by

rn ¼ffiffiffiffiffiffiffiffiffiri rop

¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiriðri þ tÞ

pð7Þ

If we only consider elastic deformation under pure bending, theneutral plane is positioned at t/2. Denoting the initial specimenlength L0, its final length L, and the distance from the neutral planeto an arbitrary pointq, the strain distribution can be calculated by[18]

eh ¼L� Lo

Lo¼ rh� rnh

rnh¼ ðrn þ yÞh� rnh

rn¼ q

rnð8Þ

If we consider plastic deformation, the neutral plane shifts, andthe strain distribution can be expressed by

eh ¼ lnLLo

� �¼ ln

rhrnh

� �¼ ln

rrn¼ ln 1þ q

rn

� �ð9Þ

The strains at the outer radius r = ro are

eo ¼ lnro

rn¼ ln

roffiffiffiffiffiffiffiffirorip ¼ ln

ffiffiffiffiffiffiffiffiffiffiro=ri

p¼ 1

2ln

ri þ tri

� �ð10Þ

The radii of curvature of die and punch are r and rp, respectively.The upper and lower radii of curvature of the die are rA and rB,respectively. At the curved part above the specimen, ri = rA and atthe curved part below the specimen ri = rBp. Applying Eq. (10), weget for the strain at the curved parts, eA = 0.156 and eB = 0.210.Since in pure bending only a bending moment acts, the specimendeforms uniformly and therefore we can calculate the strain fromthe bending force. However for dimple forming, the deformation ofthe curved part is not uniform, for an axial load (due to binderforces) and a friction force act [upon the specimen]. Therefore, inreal dimple forming a maximum strain must occur at a certainpoint. This [issue] is to be examined by FE analysis in the following.

(a) FE mesh (element type: CPE4)

(b) Deformed configuration at loaded state

Fig. 4. 2D 1-step stamping FE analysis of 1-set.

(a) Initial FE mesh

(b) Deformed configuration at loaded state

Fig. 5. 2D 1-step stamping FE analysis of 9 set.

Fig. 6. Measurement of thickness.

M. Kim et al. / Materials and Design 54 (2014) 121–129 123

3. FE model for the existing stamping process

The 1-step stamping process is modeled in ABAQUS (2010)using a specimen of length 12.5 mm and thickness 0.5 mm, width5 mm [19]. The FE mesh is composed of 4-node plane strain ele-ments (CPE4). The properties of Zircaloy-4 are elastic modulusE = 108 GPa, yield strength ro = 364 MPa, and anisotropy coeffi-cient r = 2.265 (Table 1) [20]. Punch, die and binder are assumed ri-gid, and right and left boundaries of the FE model are fixed. Thefriction coefficient is f = 0.2, the binder force BF = 20 N and the bin-der length l = 2.78 mm. Radii of curvature of die and punch are rand rp, respectively; the upper and lower radii of curvature of thedie are rA and rB (Fig. 4).

At the specimen’s upper and lower curved parts, the maximumstrains are eA = 0.262, eB = 0.351. To check the appropriateness ofthe model comprising [only] 1 set with fixed ends, the left andright boundary [in the 1 set model] are set free and results arecompared to the set in the middle of a model comprising 9 setswith free ends (Fig. 5). In both cases we find eA = 0.262 andeB = 0.351, and the thickness measurements at the points of eA

and eB (Fig. 6) shows t1 = 0.430 mm and t2 = 0.409 mm for bothmodels. Moreover, comparing the coordinates at the left boundary,no displacement can be observed (Table 2). In 1-step stamping thecurved part is formed in one step and, as in pure bending, a highdeformation occurs at the lower curved part (with low r) but com-pared to pure bending a high strain is observed. A reasonable ele-ment number is found to be ne = 1.1 � 103. Further eref = 0.348 [thestrain for the 1-step model and ne = 1.1 � 103] is taken to be thereference strain value (Table 3).

4. 2-Step stamping model

4.1. Determination of design variables for the 1st stage and design(method)

Due to the concentrated [high] deformation at the curved partin 1-step stamping, cracking may occur at the point of eB. In orderto reduce the deformation at this point, we perform a 2-step pro-cedure as illustrated in Fig. 7. The optimal shape of the 1st stamp-ing tool is the die shape with minimized emax, the maximum valueof eA and eB. Consider the manufacturability, the 1st stamping tool

Table 2Comparison of 1 set and 9 set.

# of set e values Thickness Coordinates

eA eB t1 t2 x y

1 set 0.262 0.351 0.430 0.409 6.250 1.8909 set 0.262 0.351 0.430 0.409 6.250 1.889Gap 0 0 0 0 0 0

Table 3Comparison of emax values in 1-step and 2-step stamping for different values of ne

along x-axis.

ne along x-axis 1-step 2-step

100 0.351 0.2931.1 � 103 0.348 0.315Gap (%) �1 7

Fig. 7. 2D 2-step stamping model.

Fig. 8. Process variables of 1st stamping tool.

ln

0.2 0.4 0.6 0.8 1.0

ε n

0.86

0.88

0.90

rA = 1.50 mm, 50θ =

εref = 0.348

Fig. 9. Distribution of en with the variation of ln.

124 M. Kim et al. / Materials and Design 54 (2014) 121–129

of the 2-step stamping model is composed of two circular arcs(Fig. 8). To keep the shape simple, values for rA (in mm) and h (indegree) are determined by variable minimization. The 1st stamp-ing tool leads to deformation of parts which finally regain its initialshape.

In particular, the circular arc c–f makes the middle part of theshape deform. During the second stamping stage the central partis flattened while the curved part is compressed, thereby mitigat-ing the [initial] deformation. With regard to manufacturing issues,the point of intersection of the two arcs, c, is an inflection point andnot a cusp. Point d lies on the axis of symmetry while points b, c,and d lie on a straight line. h is the angle corresponding to arc c–f, and rB can be calculated by means of rA and h. With b as the cen-ter point of the arc a–c, an arbitrary point c can be determined bymeans of the angle h in the triangle where rA is equal to the lengthof the hypotenuse bc. If point b has the coordinates (x, y), thenpoint c has the coordinates (x + rA sin h, y + rA cosh). When deter-mining rB and the length of line de (=l) from s = x + rA sinh and h,then rB = (x/sinh) + rA and l = rB cosh. The circular arc c–f is the arcwith center d, which forms with above determined values for rB

and l. Center d has the coordinates [0, y + (rA + rB)cosh] and f thecoordinates [0, y + (rA + rB)cosh–rB]. The stamp’s width decreaseswith increasing rA, while the stamping depth increases whenincreasing.

4.2. Range of design variables for 1st stamping tool and reasonableelement number

In order to determine the range of design variables, the changein strain for an arbitrary rA is observed. We find the strainemax = 0.293 to be minimum for the combination (rA,h) = (1.5 mm, 50�). Although accurate results were obtained usinga model with ne = 1.1 � 103 elements (Table 3), the model withne = 500 elements is preferred due to the – in light of lower compu-tational costs – small deviation in emax of 0.6%. (emax = 0.313)(Table 3).

4.3. Optimum binder length

Fixing (rA, h = (1.5 mm, 50�), we decrease l from 2.7 to 0.6 mm inintervals of 0.3 mm. Studying the influence of l on the strain, wefind that for l 6 0.9 mm, emax converges to 0.299 (Fig. 9). Furtheranalyses are therefore performed with l = 0.9 mm. x- and y-axesare normalized so that ln = l/lmax and en = emax/eref. Takingl = 0.9 mm in the 1-step stamping model, we obtain emax = 0.347,which corresponds to a 0.2%-difference to eref.

4.4. Influence of design variables (rA, h) on strain

Fixing l at 0.9 mm, we study the strain while changing rA from1.40 to 1.60 mm in 0.02 mm intervals and h from 48� to 52� in 1�intervals. Normalizing the x-axis such that rAn (=rA/rA|max), we finda strain minimum for h = 51� (Fig. 10(a)). Further, introducing hn

(=h/hmax) as x-coordinate, we get a minimum strain at rA = 1.58 -mm, and the corresponding strain distribution for h = 48� is theopposite of the strain distribution for h = 52� (Fig. 10(b)). As a re-sult, for (rA, h = (1.58 mm, 51�) we get an emax value of 0.295, whichis 15% smaller than eref.

4.5. Optimum value

We expressed eA, eB as a function of rA, h in the form of Eq. (11).rA lies in the range 1.40–1.60 mm, h in the range 49–51�.

ei ¼ ai þ bixþ ciyþ dixyþ eix2 þ fiy2 ð11Þ

where x = rA, y = h and ai, bi, ci, di, ei, fi (i = A, B) are constants. Valuesobtained by RSM are listed in Table 4. rA is varied in the range 1.40–1.60 mm (interval 0.01 mm) and h in the range 49–51� (interval0.1�). The corresponding strain values calculated by above equationare depicted in Fig. 11. The minimum value for Eq. (11) with

rAn

0.88 0.92 0.96 1.00

ε n

0.84

0.88

0.92

0.96

50

48θ =49

51

52

εref = 0.348

(a) Variation of εn with rAn for various θ

θn

0.92 0.94 0.96 0.98 1.00

ε n

0.84

0.88

0.92

0.96

(mm)Ar

1.44

1.48

1.52

1.56

1.60

1.40

εref = 0.348

(b) Variation of εn with θn for various rA

Fig. 10. Variation of en.

Table 4Coefficients of functions obtained by RSM.

Coefficient eA eB

a �1.563 �4.629b 1.313 �2.205c 0.032 0.276d �0.193 0.566e 0 �0.003f �0.017 0.012

Fig. 11. Response surfaces of en.

(a) FE mesh (element type: C3D8)

(b) Distribution of equivalent strain at loaded state

Fig. 12. 1-Step stamping model for 3D FE analysis.

M. Kim et al. / Materials and Design 54 (2014) 121–129 125

RA = 0.99, RB = 0.97 is determined by minimum of the intersectionline and is found to be 16% smaller than e ref (emax = 0.293).

By a 2D FE model, we determined the optimum value for the 1ststage stamping tool by minimizing the principal strain. To confirmthe real effectiveness of the shape improvement of such a 2-stepstamping model, a 3D analysis is performed. Although themaximum principal strain is identical for the 2D and 3D model,the minimum principal strain in the 2D model shows in thicknessdirection. However, since in the forming limit diagram the

minimum principal strain is in width direction, strains obtainedby the 2D model cannot be used.

Fig. 13. 2-Step stamping model for 3D FE analysis.

Fig. 14. Two ways of measuring limit strains.

Minor strain-0.4 -0.2 0.0 0.2 0.4

Maj

or st

rain

0.0

0.2

0.4

0.6

experimental data

2-step

weak point

layer average

1-step3D

uniaxial model

Fig. 15. Limit strains for two ways of strain measure.

Minor strain-0.10 -0.05 0.00

Maj

or st

rain

0.10

0.15

0.20

2-step

experimental data1-step

- 5%

t = 0.7 ~ 1.0t = 0.4 ~ 0.6

t = 0.4 ~ 1.0

Fig. 16. Limit strains for t = 0.4–1.0 mm.

126 M. Kim et al. / Materials and Design 54 (2014) 121–129

5. 3D modeling and comparison of formability

5.1. Comparison of formability using FLD

Based on the optimum values determined by the 2D model, aquarter of 3D model is made. Length, thickness, and width of thespecimen are 12.5, 0.5 and 5 mm, respectively; 8-node 3D ele-ments (C3D8) are used (Figs. 12 and 13). To compare the formabil-ity of the 1-step and 2-step stamping model, the maximumprincipal strain (emax) and the minimum principal strain (emin) atthe point of emax are plotted in an FLD (Fig. 15). The FLC (forminglimit curve) in the FLD are limit strains determined experimentally.Tests are performed with a die suggested by NUMSHEET 96 and,using ATOS and ARGUS, the experimental limit strain graph is ob-tained. According to ISO 12004-2 Gaussian curve fitting is per-formed in vicinity of the cracked area, and from the 5 pointsobtained, the lowest strain value is taken as limit strain [21]. Forthe 1-step model, we get (emax, emin) = (0.347, �0.101) and for the2-step model (emax, emin) = (0.305, �0.063) (Fig. 15). Both straincombinations at the weak point lie above the FLC. However, con-cerning the strains in the dimple, the strains in thickness directiondiffer by about 90% for a radius of curvature-to-specimen thicknessratio of 3, and attain a maximum due to bending and tension at lowradii of curvature. On the other hand, regarding the limit strain, wefind by dome stretch test that the strains in thickness direction dif-fer by about 4% for a specimen with radius of curvature-to-thick-ness ratio of 100. Hence, since the influence of the bending forcedecreases with applied tension, there must be a limit for the com-bination (emax, emin). To plot emax, emin in the FLD, we determinestrains averaged over the thickness at the weak point [/weak crosssection] (Fig. 14). The average value represents the decrease ininfluence of the bending force on the strain in thickness directionand is found to be (emax, emin) = (0.184, �0.048) for the 1-step mod-el, and (emax, emin) = (0.120, �0.029) for the 2-step model (Fig. 15).The strain for the 1-step and the 2-step model are located belowthe FLC so that nothing can be said regarding formability butstrains for the 2-step model are reduced. To assess the formabilityin terms of thickness change, we observe the strains for thick-nesses between 0.4 and 1.0 mm (interval 0.1 mm) (Fig. 16). The er-ror band (–5%) is shown, and we see that for all thicknesses thestrains for the 2-step model are below the error band. In contrast,for the 1-step model, at thicknesses t = 0.7–1.0 mm the averagestrains are above the error band (Fig. 16). At all thicknesses, themaximum strain is decreased and, in case of the 2-step stamping

1

2

3edge

middle

Fig. 17. Two ways to measure thickness.

x / l

thic

knes

s

0.40

0.45

0.50

0.55

0.60

1-step

2-step

t = 0.5 mm

weak point

(a)

x / l

0.0 0.2 0.4 0.6 0.8 1.0

0.0 0.2 0.4 0.6 0.8 1.0

thic

knes

s

0.40

0.45

0.50

0.55

0.60

1-step

2-step

t = 0.5 mm

weak point

(b)

Fig. 18. Variation of thickness at (a) edge and (b) middle.

Minor Stress (MPa)0 100 200 300 400

Maj

or S

tress

(MPa

)

680

700

720

740

760(A), (D) Hill 48 (a = 2)

(B), (E) Hosford (a = 6)

(C), (F) Hosford (a = 8)

1-step 2-step

transformation equationnecking mechanisms

t = 0.4 ~ 0.6 t = 0.4 ~ 0.6

t = 0.7 ~ 1.0

t = 0.7 ~ 1.0

Fig. 19. Limit stresses for t = 0.4–1.0 mm.

M. Kim et al. / Materials and Design 54 (2014) 121–129 127

model, formability is ensured for all thicknesses. Thus, formabilityis improved for dimple forming.

5.2. Comparison of thickness distribution

For the 1-step and 2-step stamping model, thicknesses at theedge and at the center of the specimen (Fig. 17) are measuredand depicted in Fig. 18. For the 1-step model, the thickness atthe weak point is smaller than 0.45 mm and the thickness differ-ence between center and first curved part is large, whereas forthe 2-step model a [rather] uniform thickness distribution is pro-vided, as can be seen in Fig. 18(a). Also, the thickness distributionin the center is ‘‘more even’’ [/less fluctuant] for the 2-step modelthan for the 1-step model (Fig. 18(b)).

5.3. Comparison of formability using FLSD

Position and shape of the FLC in the FLD depend on the strainpath. For the 2-step stamping model, the strain curve is not propor-tional due to the preceding forming process. To solve this issue, weapply the path-dependent FLSC (forming limit stress curve) [22].First, we substitute the strain obtained by dome stretch test inthe stress–strain conversion formula. Second, a necking machineis employed. Since the FLSC is influenced by the yield condition,the material’s anisotropy must be considered when converting.

128 M. Kim et al. / Materials and Design 54 (2014) 121–129

Here, the yield conditions of Hill 48 and Hosford 79 (a = 6, 8) areapplied.

The curves in the FLSC obtained after stress–strain conversionand substitution are the FLSC applying Hill 48 yield condition(A), the FLSC applying Hosford 79 (a = 6) yield condition (C) andthe FLSC applying Hosford 79 (a = 6) yield condition. In the FLSCobtained using the necking machine, curve D represents the FLSCfor the Hill 48 yield condition, curve E the FLSC for the Hosford79 (a = 6) yield condition and curve F the FLSC for the Hosford 79(a = 6) yield condition.

We determine the maximum and minimum principal stresses(rmax and rmin) at the weak point (Fig. 14) for specimen thick-nesses between 0.4 and 1.0 mm (interval 0.1 mm) and comparethem to the FLSCs. When comparing with curves A, B, and C, the1-step model curve lies for t = 0.6 mm above curve C, and fort = 0.7–1 mm above curve B, whereas the 2-step model curve liesfor t = 0.4–0.6 mm below curve C, for t = 0.7–0.9 mm below curveB and for t = 1.0 mm above curve B. When comparing with curvesD, E, and F, the 1-step model curve lies for t = 0.4–0.5 mm belowcurve E, for t = 0.6 mm above curves D and E and for 0.7–1 mmabove curve E. In contrast for the 2-step model, for t = 0.4–0.6 mm all curves lie below curve E, and for t = 0.7–1 mm, thecurves are above curve E (Fig. 19). For the 2-step model, themaximum stresses are lower for all thicknesses [as compared tothe 1-step model], and, assuming curve B, formability is provided.

5.4. Damage locus according to GTN model and comparison of strains

As for existing stamping models damage models were not ap-plied and thus void neither formation nor growth behavior wereinvestigated, the locus of damage occurring in the specimen duringforming could not be determined. Therefore, an analysis based onthe GTN model is performed [23]. The void volume fraction f, thedamage variable of the GTN model, can be determined from thedamage locus [24]. The yield criterion for the GTN model is givenin Eq. (12), and void formation and growth takes place accordingto Eq. (13).

U ¼ re

ry

� �2

þ 2q1f � cosh �q232

rm

ry

� �� 1� q3f �2� �

¼ 0 ð12Þ

_f ¼ _f growth þ _f nucleation ¼ ð1� f Þ _epl þ A _eple ð13Þ

In Eq. (12), ry, re, and rm denote yield stress, effective stress,and hydrostatic stress, respectively, whereas q1, q2, q3 are correc-tion coefficients. With q1 = 1.5, q2 = 1, q3 = 2.25 cracking can be pre-dicted well. When comparing fmax for the 1-step and the 2-stepmodel (Table 5), we find fmax = 0.114 for the 1-step model andfmax = 0.093 for the 2-step model. The damage value for the 2-stepmodel is 18% lower, which means that damage could be reducedwhen comparing to the 1-step model. emax at the weak point is0.371 (1-step model) and 0.311 (2-step model), which correspondsto a 16%-decrease (Table 5).

Table 5Comparison of fmax and emax values.

Stamping model fmax emax

1-Step 0.114 0.3712-Step 0.093 0.311Gap (%) �18 �16

6. Conclusions

In this study, FE analysis is performed for [only] one of severaldimples but which represents the whole stamping. A 2-step modeladded 1-step dies was suggested, aiming at a lower cracking prob-ability during dimple forming. First of all, the strains that developin the dimple curved part of the 1-step stamping model cannot beexplained by pure bending since in pure bending, thicknesses areidentical before and after bending, deformation is uniform; andforces in axial direction as well as friction are not considered.The strains in the curved part are the same for a model comprising1 set with both ends fixed and a model comprising 9 sets with bothends free so that the 1 set model was chosen due to its lower com-putational costs. For a model with a reasonable number of ele-ments, a maximum strain of emax = 0.348 was evaluated andtaken as reference value (eref). In order to decrease the concen-trated deformations occurring in the curved part during 1st stamp-ing, rA, h were taken as the design variables based upon which the1st stamping dies was designed. The range of the design variableswas determined as well as a reasonable number of elements. Fur-ther, the optimum binder length was found to be l = 0.9 mm and byvariable variation, the strains could be determined. eA and eB wereexpressed as functions of rA and h, and the so obtained strains werecompared to the reference value eref. A 16%-decrease could beachieved.

The optimum values were applied in a 3D model and formabil-ity of the stamping model was assessed. In order to do so, averagevalues of emax and emin (average of the values in one cross section)were plotted in an FLD. At thickness t = 0.5 mm and other thick-nesses, the strains were below the FLC. However, values of emax

for the 2-step model were lower than those for the 1-step model.In addition, a more uniform thickness distribution could beachieved by the 2-step model. Also the maximum stress in theFLSD evaluated at the weak point was lower for the 2-step model,as well as the damage variable according to the GTN model. We canthus conclude that the 2-step model provides an enhanced form-ability compared to the 1-step model.

Acknowledgment

The authors are grateful for the support provided by a grantfrom the Korea Research Foundation (Grant No. NRF-2012-0083476).

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