1202

y n

Figure I. Hexagonal array of holes in the perforated sheet

FORMING LIMIT DIAGRAM

I I I I I I I I

Vol.33,No. 8

n JJ~------ ---)--- ---------.L..l.l.~t:::t::t:t::t!t(.;

z Figure 2. Initial finite element mesh for 3-dimensional calculation of deformation behaviour of the perforated sheet, where the ligament strains have been obtained from the hatched element.

directions are given in Table I . U, and u. in Table I denote the prescribed displacements in the s and n directions, respectively. The apparent strains, e; and e; , are calculated using the following equations.

e; = Jn(sc lsc0)

e; = Jn(n)nA0) (I)

where n,10 and nA are the lengths ofOA before and after deformation, whereas Sco and scare the lengths of OC before and after deformation.

The strains in the ligament, e, and e n• can be obtained by averaging the strains in the hatched elements in Fig. 2. Fig. 3 shows calculated e, and en as functions of e; and e;. The strains in the ligament may be approximated by linear functions of the apparent strains as follows:

stretching direction

s-axis

n-axis

sand n axes

z-axis

de,= k,pe; + kmde;

den= kn.de; + knnde;

TABLE 1 Boundary Conditions of3D FEM

plane OO'A'A plane AA'B'B plane BB'C'C

U, = O u. = o U, = specified

u, = o u.= specified U, = O

u ,= o u . = specified U, = specified

Un!Us = 1/2, 1, 2

Traction free (Tz = 0)

(2)

plane CC'O'O

u. = o

u. = o

u. = o

Vol. 33,No. 8 FORMING LIMIT DIAGRAM 1203

0 .4 0.4

0 .2 0 .2 Gn c Gn c

c.v c.v 0 .0 ======= 0 .0

~ ~

(/)

Gs (/)

c.v c,v

- 0.2 -0.2·

(a) (b) -o.b.OO 0.05 0. 10 0.15 0. 0

-0. 4 ' 0.00 0.05 0 ·1 0 0. \ 5 0.20

Gs * Gn * Figure 3. The FEM calculated ligament strains e, (Uld e,, as a function of the apparent strains, e: and c;, respectively.

where km kns, km and k,~, are proportionality constants. In order to see if the proportionality constants remain constant when the sheet is stretched in the s and n directions at the same time, the FEM analysis and analytical calculation using the constants have been carried out at different displacement ratios.

The calculated results are shown in Fig. 4. The good agreement between the analytical solutions and the FEM results at strains less than 0.1 implies that the parameters k, k"', k,n and knn remain approximately constant in the strain range. From the geometry of the perforated sheet shown in Fig. I, it can be assumed that

(3)

The shear strains in the center of the ligament, e,.. and eu, can be neglected in the in-plane stretching. For von Mises material, the effective plastic strain increment in the center of the ligament, d£1> can be obtained from equations (2) and (3) as follows:

0 .6

0.4

0.2 c

{,.)

vi 0.0

{,.)

-0.2

- 0. 4

s 0 FEM U, /U. = 1/2 0 FEM U, /U.= 1

Un u, 2

1/2

1/2 1

2

by Eq.(2) (a) X FEMU,/U, =2 -- Calculated

-0.6"1-.-~~...,.,...,....:;...;...,..;~..:,..........;.....,....:,:~::-.-,...;...,..;~ 0 .00 0 .05 0 .10

(;"' s

0. 15 0 .20

c {,.)

vi {,.)

0.6

0.4

0.2

0.0

-0.2

-0.4 0 FEMU, / U.= 1/ 2 0 FEM U, / U,= t X FEM U, / U. = 2

Un / U. ---1/ 2

-- Colculo ted by Eq.(2) (b) -0.6 ,.....,......,..,..,..,..,..,.:,.;.....;...;:....;:,:.;.,..:,..,....~.:;:..;.~:-.-,.......;..,...,.:...t

0.00 0 .05 0 . 10 0 .15

C:"' n

0.20

Figure 4. The ligament strains, e, and e.., as a function of the apparent strains, c: and e;, at different displacement ratios (U, IU,). Symbols are FEM analysis results and lines are the results calculated using eq. (2).

1204 FORMWGL~TD~GRAM Vol. 33,No. 8

de 1 = (4)

For biaxial stretching, e;, e; and e;n can be expressed in terms of e,, em em and 1jr as follows:

de;, de: =AT

de~ de~ dexy A with A = [: :

de; ::] (5)

where 1jr is the angle between x and s axes of the ligament in Fig. I. The effective plastic strain increment of each ligament can be calculated using equations (4) and (5).

It is proposed that necking of the ligament of a perforated sheet occurs when the effective plastic strain of the ligament is equal to a critical strain, £c. There are the vertical ligament (ljr = 90°) and the diagonal ligament (ljr = ±30°) in the perforated sheet shown in Fig. 1. For biaxial stretching, the forming limit may be defined by an apparent strain state so that the larger one of the effective strains in the two ligaments at 1jr = 90 o

and 1jr = ±30° can reaches £c. Thus, FLC of the perforated sheet consists of forming limits obtained for the various biaxiality ratios. The £c may be defmed as the effective plastic strain of ligament at the maximum load of a uniaxial tension in the x direction of the perforated sheet.

The effective strain of ligament under unaxial tension is calculated by equations ( 4) and (5). Let x' andy' denote the axes in the uniaxial tension and width directions, respectively. The apparent strain increments, de;, de; and de:,, can be obtained using equation (3), where de;, de; and de~ are replaced by dex7, deY7 and de;'JI' and 1jr is the angle between x' axis and saxis of the ligament. Substitution of de;, de; and de;, into equation ( 4) gives the effective strain increment of each ligament. A critical effective plastic strain ofligament, £1, is one which gives rise to the maximum load.

Experimental Methods

A commercially available perforated sheet for dot-type shadow mask was used as the perforated sheet for this study. Fig. 5 shows the perforated sheet for dot-type shadow mask and the cross section of a hole whose size varies along the thickness direction. The average diameter ofhole was 0.214mm. The previous work [18] showed that the perforated sheet for shadow mask behaved as if it were the sheet having homogeneous holes

A .245----1 B

~--------0 . 272------~

FigureS. Scanning electron micrograph of the perforated sheet for dot-type shadow mask and the geometry of hole.

Voi.33,No. 8 FORNUNGL~TD~GRAM

~-----------120------------~

~---- 87 -----~

f--4--- 6 0 ---~

f----30~

PUNCH -50-~

1205

Figure 6. Experimental set up for stretching of perforated sheet for dot-type shadow mask. The perforated sheet is clamped between two '\_____ J.8mm brass sheets.

whose diameter was calculated by averaging hole sizes along thickness. The base metal of the sheet was Fe-36Ni Invar, whose Young's modulus, the yield stress, the plastic modulus and Poisson's ratio are 151 GPa, 232MPa, 1280MPa and 0.280 [12], respectively. Uniaxial tensile tests of the perforated sheet were performed. Tensile specimens with a gauge length of 25mm and a width of 6mm were cut from the perforated sheet at 0 o,

22.5 °, 450,67.5° and 90° to x axis. The specimens were tested at a constant cross head speed of3.4 mm min'1• In the uniaxial tension of perforated sheet, the apparent limit strain was obtained by measuring the ap­parent strain at the maximum load.

The FLC was obtained from in-plane punch stretching experiments in Fig. 6. The punch stretching experiment consists of clamping sheets over a lockbead and stretching them to failure using a 50 mm diameter cylindrical punch. The perforated sheet was clamped between two 0.8mm thick brass sheets containing a 30mm diameter hole to prevent from failure near bead. The perforated sheet specimens were SOmm in length, whose axial direction is parallel or vertical to x axis, and had different widths from 5 to SOmm. The brass sheets were 1 OOmm in length and had different widths from 60 to 1 OOmm. After stretching, the failure strains in the x andy directions (apparent) were obtained by measuring the distance between two equivalent points containing one hole near the cracked area.

0 0 00 =--------------,

-0.02

-0.0 6 Angle between x' and x axes

00000 0.0' -0.08 00000 22.5'

0.666/:l 45.0' 00000 67 .5' ... ..... 90 .0° -- c•r·=-O . sc•~·

-o. \9.oo o.o4 o.o8 * Gx•

0.12 0.16

Figure 7. 1be e: ..- e;. relation under uniaxial tension of the perfor­ated sheet for dot-type shadow mask under uniaxial tension in vari­ous directions.

0 0 2 0 -:r-----------,

0.15

-*X 0 0 0 .1 0

0

0.05

-- Calculated 00000 Experi mental

o.oo 0 30 60 90

Angle to x Axis, Deg

Figure 8. The calculated and measured apparent limit strains e.;.. during uniaxial tension of the perforated sheet.

1206 FORMlNG LIMIT DIAGRAM Vol. 33, No. 8

(a) (b)

Figure 9. Optical micrographs showing necking of(a) the vertical and (b) diagonal ligaments.

Results and Discussion

The best fitting of the calculated results in Fig. 3 by equation (2) yields k.., kns, km and k"" being -0.209, 0.428, - 0.877 and 1.677, respectively.

Under uniaxial tension, the specimen failed immediately after the maximum load, that is, the apparent uni­form elongation was almost the same as the apparent failure strain. Since the perforated sheet has 12 symmetry lines, the apparent deformation is almost isotropic and the apparent shear strain can be neglected in the uniaxial tension. Fig. 7 shows thee;.- .:;.relations under uniaxial tension in various directions, which can be best fitted to e;. = -0.5 e; .. Under uniaxial tension, the apparent limit strain in the tension direction, e;,. ~ can be calculated using equations ( 4), (5) and the relation of e;.= -0.5 e;, Fig. 8 represents the calculated and measured e;,. .as a function of angle between x' axis and x axes. This figure shows that the calculated results are in good agreement with the experimental data. For arbitrary stretching of the perforated sheet, the strain of one ligament (for example, the vertical ligament) can be higher than that of the other ligament (the diagonal ligament), and the ligament of higher strain will fail.

- 0.25

0 0 Experimental Calcula ted

-0.25

0

Figure 10. FLC of the perforated sheet for dot-type shadow mask.

/

Vol. 33,No. 8 FORMING LIMIT DIAGRAM 1207

Figs. 9(a) and (b) show neckings in the vertical and the diagonal ligaments, respectively. In general, the FLC of a sheet metal is represented by the plot of major strain as a function of minor strain. However, the FLC of perforated sheet may be represented by the plot of the limit strains in the directions of symmetric axes, x andy, in Fig. 1, because of the anisotropy of limit strain shown in Fig. 8. Fig. 10 shows the measured forming limits and the calculated FLC of the perforated sheet. In Fig. 10, circles indicate the measured data where the diagonal ligaments are necked and squares are the measured data where the vertical ligaments are necked. Since the experimental data are measured in the perforated sheets which are already necked, the calculated FLC seems to be a low bound conservative value. Figs. 8 and 10 indicate that this model gives satisfacory apparent limit strains of the perforated sheet during uniaxial tension and biaxial stretching.

Conclusion

The apparent limit strains of the perforated sheet for dot-type shadow mask under biaxial stretching has been measured. A model which can calculate the apparent limit strains of the perforated sheet has been proposed.

'-.... The FLC of the perforated sheet calculated by the model gave a reasonable lower bound solution of the form­ing limits.

'-

Acknowled~:ment

This work has been financially supported by Korea Science and Engineering Foundation through Research Center for Thin Film Fabrication and Crystal Growing of Advanced Materials, Seoul National University.

I. 2. 3. 4. 5. 6. 7. 8. 9.

10. 11. 12. 13. 14. 15.

16. 17. 18.

References

D.N. Lee and H.S. Kim, Powder Metallurgy, 35,275 (1992) H.N. Han, H.S. Kim, K.H. Oh and D.N. Lee, Powder Metallurgy, 37, 140 (1994) T. Slot and W.J. O'Donnell, J. Engineering for Industry, 93, 935 (1971) Dusan Krajcinovic and Kaushik Mallick. Int. J. Solids Structures, 29, 1837 (1992) W.J. O'Dorutell, Welding Research Council Bulletin No. 124 (1967) M.D. Bird and C.R. Steele, J. Appl. Mechanics, 59,398 (1992) W.J. O'Dorutell and J. Porowski, J. Appl. Mechanics, 40, 263 (1973) A Sawczuk. W.J. O'Dorutell and J. Porowski, Int. J. Mech. Sci., 17, 411 (1975) Fuh-kou Chen, J. Mater. Proc. Tech., 37, 175 (1993) Y. Nakamura, T . Tatenami and T. Ohata, Proc. Int. Conf. Technology of Plasticity, Kyoto, 3, 1231 (1990) T. Slot and T.R. Branca, J. Pressure Vessel Technology, 93, 220 (1974) G. Hausch and H. Warlirnont, Z. Metall., 63, 547 (1972) A Needleman, J. Appl. Mech., 39, 964 (1972) S.C. Baik, H.N. Han, K.H. Oh and D.N. Lee, Plastic yield behaviour of perforated sheets, unpublished result. D.N. Lee and Y.G. Kim, "Forming Limit Diagrams: Concepts, Methods and Applications", R.H. Wagoner, K.S. Chan and S.P. Keeler(eds.), TMS, Warrendale, 126 (1991) J.W. Kwon, D.N. Lee and I. Kim, Scripta Metall. Mater., 31,613 (1994) H. Iseki, T. Murota, K. Kato and S. Hayashi, Trans. Japan Soc. Mech. engineers (A), 55, 994 (1989) S.C. Baik, K.H. Oh and D.N. Lee, J. Mater. Proc. Tech., Submitted.