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Materials Processing Technology

E L S E V I E R Journal of Materials Processing Technology 58 (1996) 251-255

On the susceptibility to localized necking of defect-free metal sheets under biaxial stretching

K.C. Chan *, L. Gao Department of Manufacturing Engineering, Hong Kong Polytechnic University, Hung Hom, Hong Kong

Received 23 December 1994

Industrial summary

Since the introduction of the concept of the forming-limit diagram to represent acceptable limit strains during sheet metal forming by Keeler and Backofen in 1963, it has become a major research topic and a considerable number of experimental and theoretical studies have been carried out. In most of the theoretical models, a type of defect is introduced and accommodated into the theoretical formulation. However, these models have met with only partial success. Moreover, in real materials, there do exist various types of defects, and their effects on limit strains may be overlapping and interacting. A single theoretical model that can fully explain or predict the formation of localised necking in a wide range of materials seems unrealistic. In this paper, a model based on the upper-bound theorem has been proposed to assess the susceptibility to localized necking of defect-free sheets under biaxial stretching. Without assuming a defect, the effects of the work hardening, anisotropy and geometry of a sheet on the limit strains have been analyzed. The significance of the findings is discussed.

Keywords: Forming limit; Sheet metals; Work-hardening; Anisotropy; Geometry effect

I. Introduction

A considerable number of theoretical studies have been carried out since the introduction of the concept of the forming limit diagram by Keeler and Backofen [1] in 1963 to represent acceptable limit strains during sheet-metal forming. However, unless some type of defect or imper- fection is introduced and accommodated into the theoret- ical formulation, classical plasticity theories based on a smooth yield surface and the normality rule cannot predict localized necking in biaxially-stretched metal sheets [2,3]. There are, basically, three groups of imperfections pro- posed in the literature. Geometrical imperfections are classified as the first group. Whilst initial thickness non-uniformity in a sheet, as suggested by Marciniak and Kuczynski [4], is one type of geometrical imperfection, the inhomogeneous distribution of voids is also a type of geometrical imperfection [5,6]. Marciniak and Kuczyn- ski theorized that the limiting strain observed during positive stretching is a result of neck development from an initial perturbat ion in the sheet thickness. However, Tadros and Mellor [7] have argued that the model is unable to explain the physical occurrence of a rather large defect in the initial sheet if matching of theory with experiment is required. A lot of effort has also been made to correlate

*Corresponding author. Fax: +852 23625267 email: (Bitnet) MFKCCHAN@HKPCC.HKP.HK.

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the limit strains of a sheet with the nucleation and development of voids. Nevertheless, Bressan and Williams [8] have commented that voids do not play a decisive role before the onset of local necking, albeit they are extremely important in determining the subsequent strain to fracture.

The second group of imperfections is due to the inhomogeneous distribution of microstructures such as the local texture of a sheet, as proposed by Lee and Chan [9], who have shown that colonies of grains of differing textures will result in the formation of local- ized necking. The last group of imperfections may be considered as the vertex that develops on the subse- quent yield surface. St6ren and Rice [3] proposed that a vertex on the yield surface can generate a bifurcation in the state of uniform plastic deformation, which corre- sponds to the onset of local necking. However, these models merely achieved partial success. Moreover, in real materials, there do exist imperfections of various types and their effects on limit strains may be overlap- ping and interacting. Hence no constitutive equation is available to accurately depict the behavior of the mate- rials, especially for large deformation. These limitations have, in fact, inhibited the development of a general predictive model for localized necking of a metal sheet. In order to better understand the effect of the material parameters and the sheet geometry on the formation of limit strains, a model taking no account of imperfection is proposed in this paper to predict the susceptibility

252 K.C. Chan, L. Gao /Journal of Materials Processing Technology 58 (1996) 251 255

t O 2 // / , A .---

i "

/ 1o " / L, 2 Y • )

Fig. 1. A biaxially-stretched sheet with a possible neck region.

to localized necking of defect-free metal sheets under biaxial stretching.

2. T h e m o d e l

Consider a metal sheet of initial thickness to being biaxially in-plane stretched (Fig. 1). L1 and L 2 denote the lengths in the major and minor strain direction, the changes of the current thickness, t, with straining, being expressed by:

t = to e ( ': '- ,:9 (1)

where to is the initial sheet thickness. In this paper, the classical Hill's quadratic anisotropic yield criterion for an orthogonal anisotropic metal sheet is adopted, the effective stress and effective strain increment being de- termined by:

~ r 1.5 = [r I a 2 + r2 o'2 + r, r2(o" l -- 0"2) 2] (2) 1 + r2 + r, r 2

and

dg=

4 N [ r 2 ( d e 2 - r l de3)2+rl(r2 d e 3 - d g l ) 2 + ( r l d g I - - r 2 d g 2 ) 2]

(3)

where N = 2(r I + r 2 + rlr2)/3rlr2(1 + r I + r2) 2 and r 1 and r 2 are the r-values along the rolling direction and the transverse direction, respectively. For homogeneous deformation, the strain components are related to the effective strain as:

e I =

4 N [ r 2 ( r l p + p + r l ) 2 + rl(r 2 + 1 + r 2 p ) 2 + ( r I - - r 2 p ) 2]

(4)

where p is the strain ratio, represented by:

de2 rl(1 + r 2 ) c c - r l r 2 P - de) - r2(1 + rl)rlr2~: (5)

The stress ratio o~ is given by:

~r2 r2(1 + rl)p + rlr2 c~ = - - = (6)

0" l q(1 + r2) + rlr2p

The strain increments d e I and de2 and the displacement increment dL~ and dL 2 are related by:

, ['dL2 ) de2 = p "de, = , n ~ - 2 + 1 (7)

For a given displacement increment, dLj, and a given strain ratio, p, the effective strain increment can be calculated by use of Eq. (3). Assuming that the Hol- lomon equation holds, i.e.:

= kg" (8)

the corresponding plastic work increment dw as a func- tion of effective strain can be calculated by the use of the following equation:

t ~i: + dg Vk

d w = ~ dg dv = T772, .~ [ ( g + d g ) l + , , _ g, +n] (9) I -T- /t

where V= L~L2t is the current material volume under discussion. In fact, the above homogeneous displace- ment field may not necessarily be the actual displace- ment increment field, being only one of the many kinematically-admissible displacement increment fields. If it is assumed that there is a neck present in the sheet and that the major strain increment of the neck is 1% greater than that of the outside region, the strain increments inside and outside the neck, denoted by subscripts a and b respectively, are represented by:

d e a l = l n ( ~ - + 1 ) = 1.01 debl

dehl = ln(dL°l------x + \ Lo, -- A l )

de,2 = deb2 = de2 (1 O)

where 'A' is the width of the imaginary neck. Experi- mental observation shows that the width of the actual neck is of the order of the current sheet thickness. The plastic work increment, corresponding to the inhomo- geneous but kinematically admissible displacement in- crement field, can then be calculated and denoted dw'. According to the upper-bound theorem, the most likely deformation mode is associated with the least plastic work incurred. The susceptibility to local necking is therefore estimated by a parameter, M, defined as:

dw' - dw M ( l l )

dw

M will decrease as the deformation proceeds, when M > 0 homogeneous deformation predominates, whereas when M = 0 and M < O, bifurcation flow and necking will occur, respectively. Since M is a function of the effective strain, the strain ratio, the work-harden- ing parameter, the anisotropic parameter and the geo-

K.C. Chan, L. Gao /Journal of Materials Processing Technology 58 (1996) 251-255 253

5 . 4

CO

2.7 X

5-'-

~ '", \'~, n=0.2 ' \\ ", '~ r l=r2 = 1

4.5 \ ',,, '\,~ to=lmm \ ',, \',~ \ ". ,\ Ll=L2=4Omm

3.6 \ ,,, \,x \\ ",,, \',~, - - - - - p=1.o

x . . . . . p=0.8 x \ \ ,,,, \~%, - . . . . . . p=0.6

\ p =0.4

1.8 \ \ \ "'" \~ , , \~ - - - - 9=0.2 \ \ -..-% \ . . -. ,,,,,,,~,,,~, - - p=0.0

0.9 "~. ~ ~ -.'.~.....

o.o I I I ~ 1 - ~ - T - - - - - r - - z =: 0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50

Fig. 2. Variation of the M value as a function of effective strain, for different strain ratios.

metrical dimensions of the sheet, their effect on the susceptibility to localized necking of defect-free metal sheets can be explored.

3. Results and discussion

In the present study, the susceptibility to localized necking of a sheet metal with a length and width of 40 mm and a thickness of 1 mm is illustrated. In the numerical calculations, the strain increment was chosen to be 0.01. If the sheet is first assumed to be isotropic and n =0.2 , Fig. 2 shows the variation of M as a function of effective strain for different strain ratios. In general, the value of M decreases with increasing effec- tive strain. However, the M value essentially shows no sign of becoming zero except for the plane-strain condi- tion, although M approaches zero with increasing effec- tive strain. This implies that in the biaxial stretching of defect-free sheets, the deformation in homogeneous, which agrees with classical plastic theory not being able to predict localized necking of a defect-free sheet for p > 0. Fig. 3 shows the strain curves for various M values. The strains are shown to become very large when M approaches zero and the stress state is close to equi-biaxial tension. If M = 0.4 x 10 s is chosen as a reference value to assess susceptibility to localized neck- ing, the effect of the n value is shown in Fig. 4. The effect is found to be severe near plane-strain deforma- tion, but is quite small for the equi-biaxial stress state, which suggests that the limit strains in the plane-strain state are controlled more by the work-hardening capa- bility of the materials, whereas in other stress states the limit strains may be more dependent on the evolution of defects. This has been demonstrated in the experi- ments of Tadros and Mellor [10], where metal sheets such as aluminum and 70/30 brass, having higher n values, do have a greater limit strain in the plane-strain

1.60

1.40

1.00

04 ~ 0.80

0.60

i i i i i i

/ / ~ /"

1.20 / , / / / I / /

/ / ,A/f/ / / n = 0 . 2 /~// / / r l=r2 = I

,7/ / tO= lmm S / / / / LI=L2=4Omm

/ / / ___ / / M_r -~ . / / / - ~ - > , / - - - - M=0.2xl0 ~ 0.40 - / y - - / - - M=O.4x10-8

, / _ _ _ H=0.6x10-s / 0.20 / - . . . . . . SwiFt i n s t a b i l i t y

, / curve 0.00 I I I I I I I

0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60 I12

~2

Fig. 3. Calculated strain curves for different M values.

state than is found for steel, which has a relatively low n value, notwithstanding that steel sheets show greater limit strains in the equi-biaxial tension state. The greater limit strain of a steel sheet in the equi-biaxial state may be related to its surface roughness. It has been indicated that for a given strain, the surface roughness of rimming steel is much less than that of soft aluminum and soft 70/30 brass, although the aver- age grain sizes are approximately the same: thus, the difference in the evolution of surface roughness has resulted in a significant discrepancy in the limit strains. Assuming planar isotropy, Fig. 5 shows that normal anisotropy has little effect on susceptibility to localized necking in the plane-strain state, whilst a large r value tends to lower the strain level, especially near the equi-biaxial tension state. The effect of planar an- isotropy on susceptibility to localized necking is also shown in Fig. 6, the trend being in agreement with that

1.60 . . . . . . / - /

. . . . . . n=0.4 1.40 n=0.3 ~ , /

. . . . n = 0 . 2 / , / ) / 1,20 / / / / / /

- 0.80 / / CO /,/,/'/"

0.60 , '~ /" S / / / / " M=0.T~xl0 -8

l:o=Imm 0.40 - " / / Li=L2=4Omm

0.20 / / / "

0.00 I I I I I I 0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60

1/2 ~2

Fig. 4. Effect of the n-value on the susceptibility to localized necking.

254

1.60

1.20

qxl 0.80

co

0.40

/

/

0.00 0.00

K.C. Chan, L. Gao /Journal of Materials Processing Technology 58 (1996) 251 255

. . . . . . . rl=r~2=l.0 - - r l=r2'= 1.5 / / _ _ - r i=r2=2.0 /

• / / / / / "

.' / / " . / / / "

/ / / .

,,'7 / /,/ /Y /

.# /" M=0.4~ I x 10 -8 . / / ~o=Imm

/ Ll=L2=4Omm /

1.50

1.20

0.90

{,3 0.60

0.30

/ / "

0.00 0.00 I I I

0.40 0.80 1.20 1.60 1/2

g2

Fig. 5. Effect of normal anisotropy on the susceptibility to localized necking.

'- . . . . . . t~'=l 3mm ' ,-'- . . . . . . . . ' ,"

- - - t 0 = 0 . T m m , , , ~ / / - ~ , ,~ / ,~ /

, / / /-/ / ' / . /

/ , / ' " " n=0 .2 j " . / r~=r2=1 .~/' / "

/ M=O.4xlO -8 / / L1=L2=4Omm

I I 0.30 0.60 0.90 1.20 1.50

1/2 ~2

Fig. 7. Effect of the sheet thickness on the susceptibility to localized necking.

of other investigations [I 1-13]. The effect of specimen thickness on susceptibility to localized necking is illus- trated in Fig. 7, where it is seen that the thinner the sheets, the more susceptibile they are to localized neck- ing: a neck forms during a deformation process, the width of the neck will be greater for thicker sheets than for thin sheets. Since the formation of a neck requires extra plastic work per unit volume in the necking region, thinner sheets are more prone to form localized necks. The trend of these findings is consistent with published experimental results. In the present analysis, the effect of the aspect ratio and the dimensions of a sheet are considered also. Fig. 8 shows the effect of aspect ratio on the susceptibility to localized necking whilst maintaining the dimension of the sheet

(Lj x L 2 = 1600 mm 2) constant, as has been adopted in some experiments [4,10]), the effect of the dimensions of the sheet being shown in Fig. 9. It is observed that a sheet with a smaller aspect ratio (#) or dimensions is less susceptible to localized necking. In the experimen- tal in-plane stretching of sheets, the # value in the plane-strain state is smaller than it is in equi-biaxial stretching and the dimension of the sheet may vary from experiment to experiment. The classical M - K model has not taken this into consideration and so its prediction tends to underestimate the limit strain near to the plane-strain state. The effect of the aspect ratio and dimensions of specimen may therefore partly ex- plain the discrepancy of the M - K prediction.

1.60

1.20

(N - 0.80

0.40

, i

. . . . . . r t = l , r2=1.5 _ _ _ r 1 = 1 . 5 , r 2 = l ..........

.." / .-" /

/ ' / / ..~"

,,'~/ ,,~/

,,,'/ ,'7 n=0.2

,2 ,,# M=0.4xl0 -8 to=Imm Ll=L2=4Omm

0 . 0 0 I I I 0 . 0 0 0.40 0.80 1.20

21 /2

\ \

,60

cO

1.50

1.20

0.90

0.60

0.30

0.00 0.00 0.30

i

# =0.6 - - I - - - - -~. . #:,o

...... #=1.4 ~ ' ./i~'"

//.// / / ' / ' /

, / / / / / / =0.2 _ z / / H=0.4x10-8

/ r1=r2=1 /" / to=Imm

/ "

I I I 0.60 0.90 1.20

1/2 ~2

/-/

1.50

Fig. 6. Effect of planar anisotropy on the susceptibility to localized Fig. 8. Effect of the aspect ratio on the susceptibility to localized necking, necking.

K.C. Chan, L. Gao / Journal of Materials Processing Technology 58 (1996) 251-255 255

1.60

1.40

1.20

1.00

--~ _ 0.80

0.60

0.40

0.20

0.00 0.00 0.20 0.40 0.60 0.80 1.00

1/2 ~2

/ ~ / " / / . - " ; /

/ ~ ' " / / / /

/ / / n=0.2 / / . " r l=r2=l

~" , / / / "" Lf °L 2 30ram , / - - Ll=L2=4Omm

/ / ...... Ll=L2=SOmm I I t I I L I

1.20 1.40 1.60

Fig. 9. Effect of the sheet dimensions on the susceptibility to localized necking.

4. Conclusions

A model based on the upper-bound theorem has been proposed to assess the susceptibility to local neck- ing of defect-free metal sheets under biaxial stretching. The effect of the work hardening, the anisotropy and the geometry of a sheet on the limit strains have been

analyzed. The effect of work hardening on the suscepti- bility to localized necking of a defect-free sheet near the equi-biaxial tension state is found to be small, the susceptibility being related more to the evolution of defects. A thicker sheet is shown to be less susceptible to localized necking. It is also established that a sheet with a small surface area is less susceptible to localized necking. The trends of the present findings are consis- tent with those in the literature.

References

[1] S.P. Keeler and A. Backofen, Metall. Trans. A, 56 (1963) 25. [2] A. Needleman and N. Triantafyllidis, J. Eng, Mater. Technol.

100 (1978) 164. [3] S. St6ren and J.R. Rice, J. Mech. Phys. Solids, 23 (1975) 421. [4] Z. Marciniak, K. Kuczynski and T. Polora, Int. J. Mech. Sci., 15

(1973) 789. [5] J.H. Schmitt and J.M. Jalinier, Acta rnetall., 30 (1982) 1789. [6] F. Barlat and J.M. Jalinier, J. Mater. Sci. 20 (1985) 3385. [7] A.K. Tadros and P.B. Mellor, Int. J. Mech. Sci., 17(1975) 203. [8] J.D. Bressan and J.A. Williams, Int. J. Mech. Sci., 25(1983) 155. [9] W.B. Lee and K.C. Chan, Proc. 3rd Int. Conf. on Technology of

Plasticity, Kyoto, Japan., July 1990, p. 1285. [10] A.K. Tadros and P.B. Mellor, Int. J. Mech. Sci., 20 (1978) 121. [11] K.S. Chan, Metall. Trans., 1614 (1985) 629. [12] W. Choi, P.P. Gillis and S.E. Jones, Metall. Trans., 20A (1989)

1975. [13] A.S. Korhonen, J. Eng. Mater. Technol. 100 (1978) 303.