Characterization and modeling of the mechanical behavior and formability of a 2008-T4 sheet sample

Int. J. Mech. Sci. Vol. 31, No. 7, pp. 549 563, 1989

002~7403/89 S3.00+.00 Pergamon Press plc

Printed in Great Britain.

CHARACTERIZATION AND MODELING OF THE MECHANICAL BEHAVIOR AND FORMABILITY OF A 2008-T4 SHEET SAMPLE DANIEL J. LEGE, FREDERIC BARLAT a n d JOHN C. BREM Aluminum Company of America, Alcoa Laboratories, Alcoa Center, PA 15069, U.S.A. (Received 21 October 1988; and in revised form 13 March 1989) Abstract A 2008-T4 sheet sample has been characterized and its mechanical behavior and formability have been modeled. Uniaxial tensile and equal biaxial tensile stress--strain data, compressive yield strengths, crystallographic texture, earing and the forming limit curve were experimentally determined. Bulge test specimen shape and thickness profiles were also measured after various amounts of biaxial strain. A recently developed phenomenological constitutive model of anisotropic mechanical behavior was used to predict the directionality of strength, plastic strain ratio (R) and shear strain ratio (F) values. In addition, this model was used to predict the forming limit curve for this sample. Predictions made with the recent model generally compare favorably with e.~perimental results and with predictions made using the Taylor/Bishop and Hill theory. According to the data obtained in hydraulic bulge testing, the 2008-T4 exhibited apparent isotropic behavior. However, in cup drawing another axisymmetric deformation mode this material exhibited anisotropic behavior, as indicated by the formation of ears and troughs.

NOTATION constants in the saturation stress equation (equation 1) material coefficients in the yield surface defined by equation 3 exponent in the yield surface defined by equation 3 uniform elongation measured in a tensile test eu K t , K2 stress expressions for yield surface defined by equation 3 (invariant for the isotropic case) k stress coefficient in Ludwik equation (equation 2) strain hardening exponent in Ludwik equation (equation 2) n R plastic strain ratio g, true strain true strain rate F shear strain ratio o ~ true stress

A,B,C a, h, p m

subscripts 1,2,3

denote principal directions of loading denotes the rolling direction in the sheet Y denotes the transverse direction in the sheet 0 denotes the angle between the loading and sheet rolling directions

X

INTRODUCTION

Mathematical models ofcommercial forming operations are being developed at a fast pace. While improved process understanding and computing capabilities are making these models more accurate and sophisticated, the limiting condition may become the material description. A recently developed phenomenological yield function, the Taylor [1 ]/Bishop and Hill [2] theory for polycrystals, and experimental test results are used in this work to characterize the mechanical behavior of a 2008-T4 automotive sheet sample. The models are used to predict directionality of properties and forming limits, and the predictions are critically compared to experimental results to assess applicability to forming process models. MATERIAL

Alloy 2008 was specifically developed for automotive applications that require good formability. This material is typically formed in the solution heat-treated and naturally aged 549

550

DANIEL J. LFGF et al.

temper (-T4). After forming, parts are solution heat-treated and artificially aged (-T62 temper) to develop required strength levels. A 1.24 mm thick sample of 2008-T4 sheet was characterized in this work. The sample had the following composition as determined by remelt analysis: Si 0.605

Fe 0.13

Cu 0.93

Mn 0.06

Mg 0.395.

The photomicrographs shown in Fig. 1 illustrate the AI12Fe3Si constituent particle distribution and grain structure of this 2008-T4 sample, while the (200) and (111) pole figures determined by X-ray techniques and given in Fig. 2 characterize its crystallographic texture. The average ASTM grain size, as determined by standard methods, was found to be # 6. PROCEDURE

AND RESULTS

Uniaxial tensile properties of the 2008-T4 sheet were determined from specimens taken at 0, 15, 30, 45, 60, 75 and 9 0 to the rolling direction. Standard tests were conducted to determine 0.2% offset yield strength (YS), ultimate tensile strength CUTS) and total elongation in a 50 m m gage length (ELONG). These tests were run at a crosshead speed of 2.5 mm/min to YS and at approximately 20 mm/min beyond YS. Full stress-strain curves were determined with duplicate tests in the seven sheet orientations at a constant crosshead speed of 2.5 mm/min. From these tests, the constants of the Ludwik stress strain equation Ca = ke") and uniform elongation (eu) were determined. Individual measures of YS, K, n and eu varied by less than 1.5 % of the respective average values. Extensometers measured strains in both the pulling (~11) and width (e22) directions of the specimens throughout the constant crosshead speed tests. The plastic strain ratio Cdefined experimentally as R = ~22/e33 ) was derived from the slope of the true width strain-true length strain curve evaluated from the length strains of 0.01 to e u assuming volume constancy in the specimen gage section. The individual R values determined with this procedure varied by less than 3.5% of the average for any given orientation. All of the data determined from these uniaxial tensile tests are shown in Table 1. A square grid Capproximately 0.75 m m x 0.75 mm) was applied to each tensile specimen using a photo-resist technique. After testing, deformed grids in uniformly deformed regions of the specimens were analyzed with a View Engineering 1220 video measurement system to determine both the plastic strain ratio (R = e22/~33) and the shear strain ratio (F = £12/el 1 )" The R and F data calculated from the grid measurements are included in Table 1. The grids were not analyzed prior to deformation because they were assumed to be invariable. This was found to be an erroneous assumption, however, and it resulted in some uncertainty in the parameters calculated from the grid measurements. Nevertheless, the R values determined at a single point along the tensile specimen strain path with this suspect procedure showed the same directionality trends as those determined with extensometer measurements over the entire strain path. The R values calculated from grid measurements were always less than those calculated from the procedure using extensometers. Individual R values determined by either procedure varied by less than 12.5 % of the average value for any given orientation, so data from both procedures are reported. However, the fact that the grids were not measured prior to deformation resulted in more questionable F value calculations, as will be discussed later. The compressive yield strength (CYS) was determined in accordance to ASTM E9, Standard Methods of Compression Testing, with specimens taken at 0, 15, 30, 45, 60, 75 and 90 ° to the rolling direction. A compression testing jig was used to prevent buckling in the specimens without interfering with their axial deformation. The CYS data are listed in Table 1. Hydraulic bulge tests were also conducted for this 2008-T4 sheet sample. The test equipment and procedures for calculating stress strain data are described by Young et al. [3]. The bulge specimen diameter is 175.4 mm. Specimens were deformed at constant true thickness strain rates of 0.0005 s - ~, 0.005 s - ~ and 0.05 s - 1. True membrane stress (a)-true thickness strain (e) data were calculated for each of these tests and the data were fit to both a

Mechanical behavior and formability of 2008-T4

FIG. 1. Microstructure of 2008-T4 sheet: (a) All zFe3Si constituent particle distribution: (b) grain structure.

551

Mechanical behavior and formability of 2008-T4

(200) RD

553

(111) RD

45///~~315

To/ 9O

........

45~~-~~315 o

135

~//225

o

135 18~0~ 225

a

b

FIG. 2. Pole figures determined for 2008-T4 sheet: (a) (200); (b) (111).

Voce-type saturation stress equation (equation 1) and the Ludwik equation (equation 2): = A - B e x p ( - Ce) a = ke".

(1) (2)

The constants determined for the best fits to these equations, as well as maximum stresses and strains for each bulge test, are shown in Table 2. Other bulge tests were conducted at the 0.005 s - 1 strain rate, but they were terminated near true thickness strains of 0.10, 0.20, 0.30 or 0.40 at the specimen pole so that shape profiles could be determined. In a given specimen, the profiles for different sections containing the pole and a material direction (0, 45, 90 ° to the rolling direction) were measured. For each section, the two coordinates corresponding to a point on the outer surface of the specimen were determined using a Cordax 1808-M point contact coordinate measuring system. Each coordinate was obtained with an accuracy of +0.01 m m ( + 0.0003 inch). Approximately 60 points per section were measured, including 15 points over a length of 25.4 m m (1 inch) near the pole of the bulge specimen. These data are illustrated in Fig. 3. The radius of curvature was calculated within this area. Table 3 presents these radii for the three sections from each of the four specimens evaluated. It is interesting to note that no significant difference exists between the radii of curvature of different sections of the same sample. Bulge specimen thickness profiles were also determined from the tests that were terminated near strains of 0.10, 0.20, 0.30 and 0.40 with the Cordax 1808-M measuring system. The profile for the test terminated near 0.10 strain is shown in Fig. 4 as an example. All of the thickness profiles that were determined indicate that equal biaxial stretching occurs in the bulge test. X-ray techniques were used to determine the crystallographic texture of the 2008-T4 sheet sample. (200) and (111) pole figures (Fig. 2) were used to calculate the crystallographic orientation distribution function (CODF) of this material. C O D F data were also calculated for material deformed biaxially in bulge tests to thickness strains near 0.10 and 0.20. (Standard X-ray techniques could not be used for pole figure analyses on specimens with larger thickness strains owing to excessive curvature at the poles.) No significant differences were observed between the C O D F data determined for the sheet sample and the bulge specimen deformed to a strain near 0.10. Only slight differences in texture were observed between the sheet and the bulge specimen deformed to a strain near 0.20. The experimental texture data for the specimen deformed biaxially to a thickness strain of approximately 0.10

Tensile yield strength (MPa)

156.5 156.0 154.0 148.0 145.5 143.5 141.5

Test direction (o to RD)

0 15 30 45 60 75 90

270.5 271.0 271.5 265.5 261.5 260.5 258.0

Ultimate tensile strength (MPa) 29.0 26.5 27.5 26.0 29.0 28.5 27.5

Total (%) 22.9 23.3 23.2 24.5 24.4 23.5 23.8

Uniform {%)

Elongation in 50 mm

477.1 484.1 484.4 473.7 463.7 463.8 462.7

k (MPa)

o- = ke"

0.232 0.235 0.234 0.238 0.238 0.244 0.246

n

T A B L E 1. U N I A X I A L T E N S I L E A N D C O M P R E S S I V E T E S T DATA F O R A 2 0 0 8 - T 4

(1.24 m m

0.878 0.816 0.626 0.498 0.499 0.512 0.534

Extensometer 0.684 0.736 0.567 0.400 0.418 0.422 0.446

Grid

Plastic strain ratio R = /:22 g,33

SHEET SAMPLE

Shear strain ratio,

0.0213 0.0032 0.0185 0.0108 0.0460 0.0118 0.0016

F = gl2/~ll

THICK}

143.5 147.0 152.5 154.5 149.5 148.0 144.0

Compressive yield strength (MPa)

,,..-

rr~ e~

r-

Z

Mechanical behavior and formability of 2008-T4

555

TABLE 2. EQUAL BIAXIALBULGE DATAFOR A 2008-T4 SHEET (1.24 mm) SAMPLE o- = A - B ( e x p - Ce) Strain rate (s- 1)

A (MPa)

B (MPa)

0.0005 0.005 0.005 0.05 0.05

447 429 427 419 408

248 234 228 227 233

Maxima

~ = ke"

k (MPa)

n

Stress (MPa)

Strain

513 497 493 485 490

0.253 0.237 0.236 0.230 0.238

415 410 406 403 394

0.465 0.497 0.485 0.497 0.449

4.30 5.06 4.91 5.24 6.14

(a) Stress-strain equation fits are valid for strains from 0.05 to maximum strain.

T A B L E 3. R A D I U S O F C U R V A T U R E M E A S U R E M E N T S AT T H E P O L E O F B U L G E S P E C I M E N S

Radius of curvature (mm) for an arc at the following directions with respect to the RD Thickness strain at specimen pole 0.08 0.18 0.28 0.37

(0.10) (0.20) (0.30) (0.40)

0°

45 °

90 ~'

170.9 124.3 106.0 94.1

171.2 124.1 106.8 93.7

172.4 124.2 106.4 93.4

GA2~,19

50

/

'

'

' ---~--' ~ - - - - - - ~ .s~*'~ . . . . . . . . . . . .

"-- ~

e,

o -100

-75

-50

-25

Distance

0

' ' I ~ Near 0.3 I "~_~.~Near 0.4 /

25

from

pole,

50

75

100

mm

FIG. 3. Experimental bulge profiles of 2008-T4 sheet for different values of the thickness strain at the pole of the bulge specimen.

0.20

i

0.15

-

~) 0 . 1 0

-

i

2008-T4 i

I

(Sheet) i

I

i

.......

0o

- ___

45 ° 90 °

C ,B

~)

1-

I,,-

i 0.05

"

0.00 -100

I

I

I

I

I

I

[

-75

-50

-25

0

25

50

75

Distance

from

pole,

100

mm

FIG. 4. Experimental thickness strain profiles for 2008-T4 sheet measured along directions 0, 45, and 90 ° from the rolling direction.

DANIEL J. LEGE et al.

556

was separated into four major components modeled as a Gaussian distribution plus a random component. The data determined with this analysis are shown in Table 4. Earing was measured for this 2008-T4 sheet on cups that were drawn 40%. The blank and punch diameters used to draw the cups were 162 mm and 97 mm, respectively, and the punch radius was 4.75 mm. Quaker 81X was used for lubrication. Nine cups were drawn in this fashion and percent earing was calculated as follows: % earing =

average p e a k - average valley x 100%. average peak

The average earing from these nine measurements was 3.9%, with a standard deviation of 0.11%. The cup profile was also measured on one of the cups with the Cordax 1808-M measuring system. The forming limit curve was experimentally determined according to a procedure developed by Hecker [4] to assess the stretchability of the 2008-T4 sheet. Specimens (152.4 mm long and either 50.8 mm, 108 mm, or 152.4 mm wide) were sheared from the sheet, vapor degreased, masked, deoxidized, heated in sodium silicate solution and photoresist gridded with a 2.54 mm diameter circle grid. Three lubrication conditions (dry,

TABLE 4. MAJOR COMPONENTS OF THE 2 0 0 8 - T 4 SHEET TEXTURE. ( H K L ) [ U V W ] MODELED AS GAUSSIAN DISTRIBUTIONS: G(oJ) = G(0) exp ( - o)2,,'o9o 2)

1" 2 3 4

H

K

L

U

1,'

W

o90

Vol%

-0.017 -0.360 -0.192 -0.253

0.966 0,550 0.778 0.734

0.017 0.090 0.030 0.014

-0.017 0.494 -0.559 0.010

-0.017 -0.349 0.149 0.021

0.966 0.157 0.291 0.968

12.12 11.50 13.24 7.04

24.8 29.8 15.3 3.8 26.3

random *Cube

TABLE 5. STRAIN MEASUREMENTS FROM FORMING LIMIT TESTS FOR A 2 0 0 8 - T 4 SHEET SAMPLE (1.24 m m THICK)

True strain near failure

Specimen size (mm × mm)

Lubrication condition

152.4 × 50.8

.......................................................

Major

Minor

Dry

0,272 0.273 0.281

- 0.074 0.072 -0.065

1 5 2 . 4 × 108

Dry

0.210 0.210 0.218 0.218 0.220

- 0.003 -0.009 0.010 -0.009 -0.009

152.4 × 152.4

Dry

0.218 0.222 0.227

0.041 0.052 0.044

HalLo + P o l y e t h y l e n e

0.247 0.255 0,273

0.072 0.088 0.073

Quaker + Polyethylene

0.284 0.286 0.293 0.297

0.145 0.171 0.153 0.141

Mechanical behavior and formabilityof 2008-T4

557

N A L C O 486+polyethylene sheet, and Quaker 289+polyethylene sheet) were used in combination with the specimen widths to develop a wide variety of strain states. Tests were conducted in an 866.02 MTS machine using a clamping force of 311 kN and a punch travel of 254 mm/min. After deforming to failure, strains were measured from the grids near the fractured area with a video-camera/computer grid strain analyzer system. The forming limit major and minor strains are listed in Table 5. MODELING THE CONSTITUTIVE BEHAVIOR OF 2008-T4 Two models were used to describe the behavior of the 2008-T4 sample. The first one is based on the Taylor [1]/Bishop and Hill I-2] theory (TBH) for the deformation of polycrystals. In this first model, tricomponent yield surfaces for plane stress states are generated using the procedure developed by Barlat [5] and Barlat and Richmond I-6]. This model takes the crystallographic orientation distribution function (CODF) of the material as an input. The second model is a phenomenological description of the tricomponent plane stress yield surface proposed by Barlat and Lian [7]: a l K l + K21m + a l K 1 - - K 2 1 m + ( 2 - a ) 1 2 K 2 ] m= 26 m

(3)

K 1 = (axx + ha;y)~2

K 2 = {[(0"xx- htrry)/2] 2 + [pOxr]2} 0'5, where a, h, p and m are material coefficients and x and y represent rolling and transverse directions in the sheet, respectively. Strain rates (exx, eyy, kxy)are readily determined using the classical associated flow rule. This equation is a generalization of Hershey's [8] or Hosford's [9] isotropic criterion, but it is restricted to plane stress conditions. Although this criterion is phenomenological, Hershey and Hosford found that it gives a rather good approximation of yield surfaces calculated with polycrystalline models. Logan and Hosford [10] showed that using an exponent of m = 8 gives a good approximation for the behavior of FCC metals. In equation (3), the parameter m is first set equal to 8 as recommended by Logan and Hosford. This parameter reflects the severity of the texture I-7]; for 2008-T4, which is not strongly textured, this choice is justified. Then, a, h and p are calculated from the R values measured from uniaxial tension tests conducted at 0, 45 and 90 ° to the rolling direction [7]. Yield surfaces Figures 5 and 6 show the tricomponent plane stress yield surfaces calculated with the TBH model and with equation (3), respectively. The average R values from the two experimental procedures were used for the latter calculation:

R 0 = 0.78,

R4s = 0.45,

R90 =

0.49.

Coefficients for equation (3) were calculated according to the methods described in detail in Ref. [7], and the results follow: a=1.24,

h=1.15,

p=1.02.

Both yield surfaces (Figs 5 and 6) are very similar, although a very different approach was taken to determine each of them. In the following sections, these yield surfaces are used to predict mechanical properties for different loading conditions. Yield stress The variation of tensile yield stress according to the angle 0 is illustrated in Fig. 7(a). Yield stresses shown in this figure have been normalized by the yield stress determined in the rolling direction. Both the TBH model and equation (3) predict a decrease of the yield stress with an increase of 0 for 0 > 25 °, and both models appear to adequately predict the trend of the experimental tensile yield strength data. The experimental compressive yield strength data shown in Fig. 7(b) do not exhibit the same directionality as the experimental tensile data or the model predictions. The maximum compressive yield stress occurs at 45 ° to the rolling direction, while the minimum occurs parallel to the rolling direction. MS

31:7-F

558

DANIEL J. LEGE e t

1.5

O ke,,. °~ (/I

al.

I

1

I

I

1.0 0.5

"O O

N

0.0

E

s

0 Z

-0.5

S = 0.3" S=0.2 S=0

/

I

-1.0 -1.1

-0.5 0.0 0°5 1.0 Normalizedstressin RD

1.5

FIG. 5. Taylor/Bishop and Hill (TBH) tricomponent plane stress yield surfaces for 2008-T4 sheet. Stresses are normalized by the uniaxial tensile stress in the rolling direction (#). Solid lines represent the intersection of the surface by a plane S = axr/,'i parallel to the plane (axx/5 cr/6) and projected onto it. Dashed lines represent the projection of the points whose strain ratio kyy/~xxis constant. For more details see Refs [4, 5].

1.5

Q k-

I

[

I

1.0

C °D

0.5

~

"O

O .N n

0.0 s = o.ss

E O

~'

S=0.5 S=0.4

Z

-0.5

S = 0.3 S=0.2

S=0 / -1.0 -1.1

1

I -0.5

0.0

0.5

Normalized

stress

I 1.0

1.5

in RD

FIG. 6. Tricomponent plane stress yield surface of 2008-T4 using equation (3).

R values R values were calculated using the following definition:

R o = ~22/~33

(4)

where directions 2 a n d 3 respectively represent the width a n d thickness directions in a tension specimen. ( N o t e that for theoretical p r e d i c t i o n s of R value, strain rates are used r a t h e r t h a n strains because strain rates are derived from the yield surface. F o r linear strain p a t h s a n d m o d e r a t e strains, the two definitions of R are equivalent.) F i g u r e 8 shows this p a r a m e t e r as a function of the angle between rolling a n d tension directions, 0. T h e p h e n o m e n o l o g i c a l m o d e l gives a g o o d a p p r o x i m a t i o n of R value v a r i a t i o n (dashed line). This is n o t surprising since the coefficients of e q u a t i o n (3) are t a k e n so as to m a t c h Ro, R45,

Mechanical behavior and formabilityof 2008-T4 1.2

I

I

I

I

r

I

I

- - "

~

170

~

~7--

I

• Tension • Compression

Th. T B H Th. EQ. 3

~

1.1

I

160~,..

=^~

C) Exp.

559 I

I

I

I

~, \

N 1.or

~)150

._. 0.9

0.8

!

-

I

I

I

0

10

20

1_

30

_2__._~___

40

50

~

140

,

I

I

I

60

70

80

Angle b e t w e e n rolling and tension a

130

90

directions

I t I i 1 ~--__ 20 30 40 50 60 70 80 90 Angle between rolling and loading directions ~ 10

b

FIG. 7. (a) Experimental (A) and theoretical [ ........ TBH, equation (3)] tensile yield stress variation with test direction for 2008-T4 sheet. Stresses are normalized by the uniaxial yield stress in the rolling direction. (b) Experimental tensile and compressive yield stress variation with test direction for 2008-T4 sheet.

2.0 ~ ~ - - - - T -

i

I

r

I

I

[ |

Th, TBH

~

=~ ~

1.5 ~ -

m

•

Th. EQ. 3

Exp.

o Exp,(Grids> j

Ilc

°"°I- 2_ 0.5

0.0

~ 0 Angle

10

20 between

30

~ 40

rolling

~

50 and

60 tension

70

80

90

directions

FIo. 8. Predicted R value variation with test direction by the TBH and equation (3) models together with experimental data obtained by extensometer (A) or grid (O) measurements.

and R90. The TBH model (solid line) gives the good trend, a decrease of R with an increase of angle. However, the amplitude of the variation is too large. This result is not surprising because it is well known that this theory does not provide a good prediction of the R parameter for the case of FCC metals. This plot also shows that there can be some variability in experimental R value measurements depending on measurement technique. F value

This parameter was introduced by Barlat and Richmond [6] to characterize shear during uniaxial tension. F is theoretically related to earing characteristics [6-]. It is defined by: F0 = ~12/~tl,

(5)

where direction 1 is the tension direction. The variation of F with respect to test direction is represented in Fig. 9 for both constitutive models. An attempt was made to evaluate this parameter experimentally using an image analyzer which measured the geometrical features of the grids applied on the tension specimens. However, although a careful procedure was followed to obtain the difference between grid angles measured in the grip ends (un-

deformed) and those measured in the uniformly deformed region of the tensile specimens, the

560

DANIEL J. LEGE et al.

0.10

I

i

I

i

i

i

i

]

f

._=

/-

o=

o

\

O.O5

== _o ._=

0.00

°

0.05 J=

-0.10

Th. TBH

- m

!

J

I

I

~

~

--

~

Th. EQ. 3

i

Exp. (Grids)

!

•

t

I

0 10 20 30 40 50 60 70 80 90 Angle between rolling and tension directions

FIG. 9. E x p e r i m e n t a l (O) a n d t h e o r e t i c a l [ T B H , - -- - e q u a t i o n (3)] v a r i a t i o n of F (iq 2/i:~ ~ ) with test d i r e c t i o n for 2 0 0 8 - T 4 sheet.

discrepancy between experimental values is too large. The precision of the angle measurements is not better than _+0.3 °, which is very large compared to the order of magnitude expected for this parameter (between 0 and 2°). Better accuracy may be obtained if the same grids are measured before and after deformation. Although a general trend seems to appear, F being a maximum for tests conducted at approximately 60 ~' to the rolling direction, it is difficult to come to a conclusion concerning the ability of the models to predict this parameter. Bulge test Hydraulic bulge tests were conducted for several reasons. First, the experimental specimen geometries after deformation (Figs 3 and 4) can be compared to shapes predicted by FEM codes. (These comparisons will not be discussed in this report.) This test also provides a stress-strain curve for strains larger than those obtained in uniaxial tension. This cr-~ curve is used for forming limit calculations. The last reason for conducting bulge tests is to estimate the effect of material anisotropy on an axisymmetric deformation process. Actually, profiles like those reported in Figs 3 or 4 were measured in three different sections of the sample, each of them containing the sheet normal direction and either one of the rolling, transverse or 45 ° directions. However, no significant differences were observed between results obtained in any section. This suggests that equal strains develop in the plane of the sheet at the pole of the specimen during bulge testing. Spherically shaped bulge specimen profiles for anisotropic sheet metals have been reported by several investigators [11-14] in agreement with this work. However, none have proven rigorously that this result implies that a condition of equal stress occurs at the pole of the bulge specimen [ 15, 16]. The behavior observed on 2008-T4 sheet is also consistent with a finite element analysis of hydrostatic bulging by Yang and Kim [17]. They predicted that no directionality should occur in radii of curvature as well as in strains at the pole of bulge specimens.Hill's [18] planar anisotropic yield function was used in the FEM analysis. This predicted result matches the experimental results obtained for the 2008-T4 sheet in this program. In addition, crystallographic texture measured at the pole of the bulge samples after 0.1 and 0.2 thickness strains indicates that no major texture change occurs during this biaxial deformation. This result is encouraging because it provides support, at this moderate strain level, for the isotropic hardening assumptions in the constitutive model. Forminq limits The experimentally determined forming limits for the 2008-T4 sheet are compared to predicted limits in Fig. 10. The Marciniak Kuczynski (M-K) [-19] and Hill's [20] analyses were used in the stretching and the drawing modes, respectively, for the predicted limits. The stretching range is the most interesting since it has been shown that in this case, forming limits strongly depend on the yield surface shape [5]. M - K analysis was used with an

Mechanical behavior and formabilityof 2008-T4 0.6 [

T

]

2000-T4 (Sheet) i E

561

i

EQ. 3 -- --TBH © Exp.

i

0.4 ! .E

\

2

f

\

"~ ~ 0.3 J~

'g

i

0.2 r0.1

0.0

[I -0.2

1 -0.1

[ I 0.0 0.1 0.2 Minor strain

I 0.3

0.4

FlG. 10. Experimental(O) and theoretical [- - - TBH, - ..... equation (3)] forminglimit diagram for 2008-T4 sheet. imperfection factor of 0.996 which can physically represent certain types of imperfections in sheet metals [21]. Predictions made using the TBH and recently developed phenomenological constitutive models are both very near those determined experimentally. Near the balanced biaxial stretching region, where the two models predict significantly different limit strains, no experimental data were obtained with the standard forming limit diagram procedure. However, grid measurements made on hydraulic bulge specimens indicate that major strain ~ minor strain g 0.30 in this equal biaxial stress state. Therefore, a data point from the hydraulic bulge test is included in Fig. 10. This measure of biaxial limit strain is better predicted when using equation (3). Even though this bulge test data point was obtained under boundary conditions different from the other experimental data points in Fig. 10, it provides support for the phenomenological model. Grid strains were also measured on tensile specimens to get an experimental point on the left-hand side of the forming limit diagram. Both the TBH model and equation 3 overestimate this uniaxial tensile strain limit.

Earing tendency It has been shown [18, 22] that the initial velocity field of a given point around the rim of a blank being drawn into a cup is related to F (the shear strain ratio) or, equivalently, is proportional to the yield stress in the loading direction that is tangential to the blank at this point. The variation of uniaxial tensile yield strength with test direction should be related to the earing profile of a cup drawn with no blank holder force (plane stress condition), while the plane strain yield strength directionality should simulate drawing when no thickening occurs in the flange (plane strain condition). In a real drawing operation where blankholder forces are used and thickening occurs, mixed (generalized plane strain and plane stress) conditions actually occur simultaneously at different locations in the flange. Procedures have been developed to calculate the cup profiles for these mixed conditions [22]. Earing profile predictions for plane stress and mixed conditions made with the TBH and new phenomenological models are compared to experimental results in Fig. l l(a) and (b), respectively. The experimental trend is best predicted by the TBH model for mixed conditions. Experimental measurements of thickness profiles in the flange of drawn cups have shown that a stress state near plane stress is most likely to occur. Stresses in the flange are nearly compressive in nature. Therefore, the experimental variation of uniaxial compression yield stress, another plane stress state, with test direction may be the correct strength measure to compare the earing profile. As seen in Figs 7(b) and 11, the compressive yield strength directionality compares more favorably to the earing profile than the tensile yield strength directionality. This result suggests that tension and compression are not equivalent and that compression data must be used for earing prediction instead of tension data. Also, it appears

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DANIEL J. LEGE et al.

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FIG. 11. (a) Experimental( --) and theoretical earing profile predicted for 2008-T4 sheet using the TBH yield surface. Plane stress ( ) or mixed conditions (. . . . . ) assumed in the flange of the cup. (b) Experimental ( --) and theoretical earing profile predicted for 2008-T4 sheet using yield surface described by equation (3). Plane stress (--) or mixed conditions (. . . . . ) assumed in the flange of the cup.

that an anisotropic strength differential effect exists and that anisotropic work-hardening laws have to be taken into account for an accurate description of material behavior. Constitutive models used in this work are limited to isotropic work-hardening assumptions and, therefore, are not able to fully predict the earing behavior of 2008-T4. CONCLUSIONS

The results of the experimental characterization and theoretical modeling of the mechanical behavior of a 2008-T4 sheet sample lead to the following conclusions: (1) Tricomponent yield surface predictions made with the Taylor/Bishop and Hill (TBH) theory are essentially equivalent to those made with a recently developed phenomenological constitutive model. Therefore, process models which require yield surface input can use the recently developed constitutive model with no apparent loss of precision. (2) Both models evaluated in this work correctly predict experimental yield strength variations with tensile loading direction and trends in forming limit strains. (3) Even though both the recently developed phenomenological and TBH models predict similar yield surface shapes, the variation of plastic strain ratio (R = ~ 2 2 / ~ 3 3 ) with tensile loading direction is better predicted by the phenomenological equation. This is an expected result because the new constitutive equation uses R o, R45, and R9o as input and because it is well known that the TBH model overpredicts the R parameter for FCC metals. (4) Experimental techniques must be improved to get accurate measures of the shear strain ratio (F) before comparisons with model predictions can be made. (5) Crystallographic texture does not change significantly during biaxial deformation to thickness strains of 0.20 in hydraulic bulge tests. (6) Directional compressive yield strength measurements match experimentally determined earing profiles from drawn cups while directional measures of tensile yield strength do not. (7) This 2008-T4 sheet sample exhibits apparent anisotropic behavior in tension, compression, and axisymmetric cup drawing. However, in hydraulic bulge tests (another axisymmetric deformation mode) this material behaves in an apparent isotropic fashion. Reasons for this contradictory behavior are unclear at this time. The above conclusions suggest that, for better material understanding and description, improved polycrystal models of deformation which include more microstructural informa-

Mechanical behavior and formability of 2008-T4

563

tion, such as o r i e n t a t i o n of n e i g h b o r i n g g r a i n s a n d p r e c i p i t a t e s t r u c t u r e , m u s t be d e v e l o p e d . F r o m a p h e n o m e n o l o g i c a l p o i n t of view, the b e h a v i o r of 2 0 0 8 - T 4 sheet is well p r e d i c t e d in the s t r e t c h i n g r a n g e a n d n o t as well p r e d i c t e d w h e n the stress state t e n d s t o w a r d u n i a x i a l l o a d i n g o r s h e a r l o a d i n g (close to the d e e p d r a w i n g c o n d i t i o n ) . T h i s suggests t h a t e q u a t i o n (3) m u s t be m o d i f i e d such t h a t t h e yield surface s h a p e is a l t e r e d s u b s t a n t i a l l y in the s h e a r m o d e a n d v e r y slightly in the b i a x i a l range. W o r k has a l r e a d y b e e n u n d e r t a k e n in this direction. Acknowledgements--The authors would like to thank Saade Harb for the experimental determination of the

forming limit curve, Mike McCleary and George Schneider for the dimensional measurements made on the View Engineering 1220 video and Cordax 1808-M point contact measurement systems, Rich Tallarico for the microstructural characterization work, Bill Fricke for his crystallographic texture separation analysis, and Owen Richmond for stimulating discussions. REFERENCES 1. G. I. TAYLOR, Plastic strain in metals. J. Inst. Metals 62, 307 (1938). 2. J. W. F. BISHOPand R. HILL, A theory of the plastic distortion of a polycrystalline aggregate under combined stress. Phil. Mag. 42, 414, 1294 (1951). 3. R. F. YOUNG,J. E. BIRD and J. L. DUNCAN,An automated hydraulic bulge tester. J. appl. Metalworking 2, 11 (1981). 4. S. S. HECKER, A simple technique for determining forming limit curves. 7th Biennial Cong. IDDRG, Amsterdam, Netherlands (1972). 5. F. BARLAT,Crystallographic texture, anisotropic yield surfaces and forming limits of sheet metals. Mat. Sci. Engng 91, 55 (1987). 6. F. BARLATand O. RICHMOND,Prediction of tricomponent plane stress yield surfaces and associated flow and failure behavior of strongly textured FCC polycrystalline sheets. Mat. Sei. Engn# 95, 15 (1987). 7. F. BARLATand J. LIAN, Plastic behavior and stretchability of sheet metals. Part I: a yield function for orthotropic sheet under plane stress conditions. Int. J. Plasticity 5, 51 (1989). 8. A.V. HERSHEY,The plasticity of an isotropic aggregate of anisotropic face centered cubic crystal. J. appl. Mech. 76, 241 (1954). 9. W. F. HOSFORD, A generalized isotropic yield criterion. J. appl. Mech. 39, 607 (1972). 10. R. W. LOGANand W. F. HOSFORD,Upper-bound anisotropic yield locus calculations assuming ( 111 )-pencil glide. Int. J. Mech. Sci. 22, 419 (1980). 11. A. N. BRAMELEYand P. B. MELLOR, Plastic flow in stabilized sheet steel. Int. J. Mech. Sci. 8, 101 (1966). 12. A. N. BRAMELEYand P. B. MELLOR,Plastic anisotropy of titanium and zinc sheet--I. Macroscopic approach. Int. J. Mech. Sci. 10, 211 (1968). 13. A.J. RANTA-ESKOLA,Use of the hydraulic bulge test in biaxial tensile testing. Int. J. Mech. Sci. 21,457 (1979). 14. ~. VIAL, W. F. HOSFORDand R. M. CADDELL,Yield loci of anisotropic sheet metals. Int. J. Mech. Sci. 25, 899 ~983). 15. G. S. K~LLARand M. J.'HILLIER,Re-interpretation of some simple tension and bulge test data for anisotropic metals. Int. J. Mech. Sci. 14, 631 (1972). 16. P. 13. MELLORand J. CHAKRABARTY,Comment on re-interpretation of some simple tension and bulge test data for anisotropic metals. Int. J. Mech. Sci. 15, 689 (1973). 17. D. Y. YANGand Y. J. KIM, A rigid plastic finite-element formulation for the analysis of general deformation of planar anisotropic sheet metals and its applications. Int. J. Mech. Sei. 28, 825 (1986). 18. R. HILL, The Mathematical Theory of Plasticity, p. 328. Oxford University Press, London (1950). 19. Z. MARCINIAKand K. KUCZVNSKI,Limit strains in the process of stretch-forming sheet metal. Int. J. Mech. Sci. 9, 609 (1967). 20. R. HILL, A theory of the yielding and plastic flow of anisotropic metals. Proc. R. Soc. Lond. A193, 281 (1948). 21. F. BARLATand J. M. JALINIER,Formability of sheet metal with heterogeneous damage. J. Mat. Sci. 20, 3385 (1985). 22. F. BARLAT,Prediction of earing in cup drawing FCC single and polycrystal. Alcoa Laboratories Progress Report KP22-2 (1988-02-09).