A Method for Direct Measurement of Multiaxial Stress-Strain

Curves in Sheet Metal

T. FOECKE, M.A. IADICOLA, A. LIN, and S.W. BANOVIC

A method has been developed that allows measurement of stress-strain curves for sheet metal

being deformed in multiaxial tension. The strain state is imposed using a modification of the

Marciniak in-plane biaxial stretching test. Resulting stresses are measured using a modified Xray diffraction (XRD) residual stress measurement system. This system is flexible enough to

allow spatial mapping of in-plane stress and measurement of stresses at specific locations of

interest on the sample, such as developing localizations. Results are presented correlating

measurements on a thin strip of AA5182 with data from standard uniaxial tension test. Also

presented are experimentally determined curves for this material in balanced biaxial tension in

both the rolling and transverse directions.

DOI: 10.1007/s11661-006-9044-y

 The Minerals, Metals & Materials Society and ASM International 2007

I. INTRODUCTION

ACCURATE sheet metal forming die designs rely

on numerical models, which in turn rely on accurate

constitutive laws and experimental data for the material

being deformed. Of particular interest to those modeling

sheet metal forming processes using the finite element

method (FEM) is the shape of the yield locus in stress

space, and in particular how the shape, size, and

location of the locus changes with strain level. Experimental data in this area would allow modelers to

predict and use a forming limit diagram (FLD) formalism in stress space rather than strain space.[1] This would

allow the prediction of maximum allowable stresses in a

part being formed, which is a potentially more intrinsic

property than the maximum strains, which have been

shown to be highly path dependent[2] and valid only for

linear strain paths (which are seldom seen in production

sheet forming[1]).

To map out the yield locus, one must be able to detect

yield in a sample that is being loaded along some

multiaxial strain path and simultaneously determine the

stresses being imposed. The easiest path to measure is

uniaxial tension, where the ratio of principal strains

(q = e2/e1) equals –0.5, and the true stress, defined as

current load divided by current cross-sectional area, is

assumed uniform across the gage section up to the point

of localization prior to failure. Measuring the stressstrain behavior along other strain paths is much more

difficult and has been attempted in several ways. Tests

involving application of simultaneous tension/torsion or

internal pressure/torsion on tubular samples have been

used to probe the yield locus,[3,4] but this sample

geometry is not readily adaptable to flat sheet samples.

Other common methods include hydraulic bulge and

cruciform sheet testing.

Hydraulic bulge tests are able to produce near

balanced biaxial stress states via knowledge of the

applied pressure,[5] where the stress at the outer surface

is calculated using membrane theory. However, this

geometry indufces significant out-of-plane bending,

which has been shown to affect forming limits[6] and

would produce a gradient in the applied stress through

the thickness of the sheet. Additionally, in bulge testing,

the deformation varies radially, from balanced biaxial

tension at the crown to plane strain near the clamped

edge. This results in a radial thickness variation. The

stress at a point must be calculated based on the local

thickness dimension, which is calculated during the test

using the measured average strain and assuming a

constant volume in the local area of measurement.

Cruciform sheet metal samples have been deformed

using orthogonally oriented hydraulic rams,[7] producing various biaxial strain states in the center of the

cruciform arms of the sample based upon the imposed

elongations. The stress is then determined using the

measured strains in the arms as input to a finite element

analysis (FEA), or is calculated as force applied to the

arms of the sample divided by the arm cross-sectional

area (this assumes no interactions between the axes to

produce shear stresses within the range of applied

loads). Cruciform tests give good accuracy for probing

the yield locus at the point of initial yield when the yield

detection technique (such as detecting a temperature

rise[8]) senses sufficiently small plastic strains. The

sample geometry for cruciform samples is an important

factor for experimental accuracy and relevance of the

data. Some experiments have produced thinned gage

sections in the center of samples by carefully machining

a thicker plate, while others have added sacrificial

additional sheets with holes in the center, forcing

T. FOECKE, Staff Materials Scientist, M.A. IADICOLA and

S.W. BANOVIC, Staff Scientists, and A. LIN, Guest Researcher, are

with the Metallurgy Division, Department of Commence, National

Institute of Standards and Technology, Technology Administration,

Gaithersburg, MD 20899, USA. Contact e-mail: tfoecke@nist.gov

Manuscript submitted March 21, 2006.

306—VOLUME 38A, FEBRUARY 2007 U.S. GOVERNMENT WORK

NOT PROTECTED BY U.S. COPYRIGHT

METALLURGICAL AND MATERIALS TRANSACTIONS A
deformation to occur in the center of the sample. If no

thinning or reinforcing plates are used, then the available plastic strain range of most geometries is small (on

the order of 4 pct[6]), and can only probe the location of

the yield locus, and cannot directly track the evolution

of the yield surface with increasing plastic strain. A

special geometry has been developed,[8] producing a

reduced cross section in the center without thinning, and

produces results up to 5 pct plastic strain before

localization occurs within the loading sections. All

cruciform samples suffer the limitation that some

assumptions must be made in order to calculate the

applied stress at the center of the sample, either a

consititutive law in FEA,[7] assuming the magnitude of

shear stresses within the gage section, or an effective

cross section over which to divide the applied loads.[7,8]

For an unbiased determination of multiaxial constitutive relations, it is essential that no assumptions about

the form of the stress-strain relation be made in

advance. Therefore, experimental geometries where the

stress state is determined using modeling approaches are

to be avoided. To maximize applicability of this test for

manufacturing purposes, it is desirable that the sheet not

be modified in thickness prior to testing, as removing the

layer that has experienced a skin-pass rolling may

significantly alter its behavior. A new test has been

developed that uses X-ray diffraction (XRD) methods to

measure the stress being applied at any point on a

multiaxially strained sheet metal sample under load.

This method can measure the stresses and strains until

localization prior to sample failure.

II. EXPERIMENTAL PROCEDURE

Biaxial deformation states are produced in sheet

metal samples using a technique based upon the

Raghavan[6] modification of the Marciniak[9] biaxial

stretching test. This test has many advantages: the

strains are applied in plane to an as-received sheet

sample, there is no frictional contact in the region

where measurements are to be made, and the initial

sample geometry can be varied so as to produce strain

states that range from uniaxial tension (q = –0.5)

through plane strain (q = 0) to balanced biaxial

tension (q = 1) (Figure 1).

As-received 5182-0 aluminum alloy sheet with an

initial thickness of 1.0 mm was cut into 15-in.- (381-mm-)

square samples. Washers for the test were produced

from 1-mm-thick sheets of 1010 mild steel, also cut to

15-in. (381-mm) squares, by drilling and deburring a

3-in.- (76.2-mm-) diameter hole in the center (Figure 2).

Each sample was loaded in a manner described previously[10, 11] at 0.5 mm/s between hold points, where the

stress was measured with the sample under load.

The macroscopic plastic strains were measured optically as the change in size and shape of an array of

lithographically printed circles using the video system

described previously.[10] At each macroscopic plastic

strain level, the stress is determined by measuring the

change in interatomic spacing in the grains due to

the applied load, from which the elastic strains and then

the stress is calculated. It is important to note that this

technique measures two different strains in the sample.

The optical technique that tracks the change in the size

and shape of a feduciary circle on the surface provides

the macroscopic plastic strain, and the X-ray technique

described subsequently measures the elastic strain in the

grains, which with the elastic constants reveals the

applied stress.

The stresses are measured in situ during the test using

a modified XRD residual stress measuring system. This

technique measures the interatomic spacing using a

diffraction peak from a given set of atomic planes within

the top 40 lm of the thickness, in the case of aluminum.

Because the sheet is in plastic yield, the stress through

the thickness is uniformly equal to the flow stress at that

point, and the surface stress measurement equals the

flow stress. The XRD system has been mounted on the

back of the forming equipment (Figure 3), allowing the

head to reach inside the tooling and probe the center of

the sample while under load. The system is equipped

with a three-axis translation stage that moves the X-ray

head and allows stresses to be measured at specific

locations on the sheet.

The stress measurements are performed using the

standard sin2

w technique (as described in Reference 12)

with the (331) peak and Co Ka radiation for our

aluminum samples. We briefly review this technique here

for completeness and understanding. As mentioned

above, the XRD technique measures the motion of

diffraction peaks to determine the interatomic spacing

of a given set of adjacent atomic planes near the surface

of the specimen in a small volume of the material

Fig. 1—Strain space description with images of the Rahaven-shaped

specimen and matching washers.[6]

Fig. 2—Through section of Maciniak tooling with the Raghaven

specimen and washer.

METALLURGICAL AND MATERIALS TRANSACTIONS A VOLUME 38A, FEBRUARY 2007—307
(i.e., the focal volume). The governing equation used for

residual stress analysis and our in-situ stress analysis is

dw  do

do

1⁄4 ð^e33Þw 1⁄21

where the elastic lattice distortion is in terms of the

interplanar spacing in the stress-free (do) and measured

(dw) states and eij is the ij component of strain (with the

right-hand side in the X-ray source/detector coordinate

system). The right-hand side of Eq. [1] may be transformed to the surface coordinate system through a

simple rotation (w), as shown in Figure 4. Typically,

stresses are determined from these average microstrains

assuming an isotropic constitutive law:

eij 1⁄4 1 þ m

E

rij þ dij

m

E

rkk 1⁄22

where the terms –m/E and (1 + m)/E are often referred to

as the effective or average X-ray elastic constants

(XECs) S1 and 1

2S2, respectively. Because the gage area

has stress-free surfaces on top and bottom, and the sheet

thickness is quite small as compared to the length and

width of the gage section (approximately 3 pct), the

assumption is made that r33 = 0 through the thickness.

Therefore our final equation for calculation of stress

from the elastic lattice strain is

dw  do

dw

1⁄4 1 þ m

E

r11 sin2 w  m

E ðr11 þ r22Þ

þ

1 þ m

E

r13 sin 2w 1⁄23

The XRD residual stress analysis is performed by

plotting elastic lattice distortion vs sin2

w, where w is the

current tilt angle at the time of the data collection. Data

from multiple tilt angles taken at one stress level are fit

with a single line or an ellipse. Note that during

sampling, when the angle w is set at a specific value,

the beam and detector tilt are oscillated during acquisition by a small angle (b = ±5 deg) in addition to the

w angle to reduce the effect of small surface aberrations

on the peak shape. The peaks are located using a

Pearson VII fit of the upper 85 pct of the peak with

subtraction of the background radiation. From Bragg’s

law and knowledge of the initial do spacing, the lattice

distortion is calculated. The slope of the linear fit relates

directly to the first term on the right-hand side of Eq. [3],

and the intercept relates to the second term. The

elliptical fit determines the third term. Although this

would suggest that both stresses in the 11 and 22

directions could be determined with one scan, the

uncertainties in determination of the second term tend

to be too large as compared to the stresses of interest.

Therefore, we only measure the stress in the direction of

tilt. The shear determined by the elliptical fit (i.e., shear

splitting) should not exist in our experimental arrangement because we align the plane of the X-ray head tilt

with the principal axes of the deformation in the sheet.

Therefore, most (if not all) shear seen is due to

misalignments in the system or effects of texture.

Because the system measures the stress in the direction

of tilting of the X-ray head, multiple samples that are

rotated about the sheet surface normal axis are used to

measure the flow curves in many directions within a

single strain state. For each stress measurement, the

biaxial straining was interrupted and the measurement

made within 90 to 180 seconds; then straining was

continued. The strain rate used was approximately

1 · 10–4 s

)1 between measurements.

The XECs used in the measurement are not the singlecrystal values, but are effective XECs that must be

determined separately to account for crystallographic

texture, chemical and alloying effects, and other variables. The sin2

w method can be particularly difficult in

samples that exhibit an elevated degree of crystallographic texture,[13] as is typically present in rolled sheet

samples. The effective XECs[14] were measured by

straining a tapered uniaxial tensile specimen[15] in a

portable tensile stage with a calibrated load cell, while

simultaneously measuring the tensile stresses using the

X-ray system. Correction factors for the XECs were

Fig. 3—XRD head in place above the specimen and Marciniak tooling.

Fig. 4—XRD axes definition.

308—VOLUME 38A, FEBRUARY 2007 METALLURGICAL AND MATERIALS TRANSACTIONS A
calculated assuming the gage section of the specimen is

under uniaxial stress, which is measured using the load

from the load cell and the elastic strain measured

through a clip-on extensometer on the back of the

sample. This was done in both the rolling and transverse

sheet directions. The effective XECs might be expected

to change with accumulated plastic strain, because this

will change the texture of the sample.[11,16,17] For

uniaxial samples, this correction is easily determined

using the method described previously while tracking

the plastic strains using an extensometer. To determine

the effect of biaxial plastic strain, additional samples

prestrained in balanced biaxial tension to different

plastic strain levels were subsequently tested in uniaxial

tension under the X-ray system in the same manner as

the unprestrained samples. This permitted calculation of

correction factors as for the as-received and balanced

biaxial prestrained materials for rolling and transverse

directions that were applied to the subsequent stress

measurements.

Accurate measurement of the stresses in the sheet

using this X-ray technique requires careful experimental

setup and interpretation. Several sources of experimental error have been considered and examined. Alignment

is very important (ASTM E915[18]), and in this particular system the collimator to surface distance must be

accurate within approximately 200 lm to maintain

focus. This is checked before each stress measurement.

The mathematical fits of the do vs sin2

w data are

elliptical in functionality, but the application of a

balanced biaxial strain to the sheet should not produce

any shear stresses, which as described previously is the

typical cause of elliptical splitting in the data. As a

practical matter, the appearance of elliptical splitting in

our data indicates that the system is out of proper setup,

usually out of focus. The X-ray system alignment and

focus are initially checked by performing a stress

measurement on a fine powder sample of the same

composition (which is assumed to be stress free). The

uncertainty of the stress measurements presented here is

based on the first standard deviation of the linear fits of

the do vs sin2

w data. This results in a typical uncertainty

for the stress measurements presented here of 8.7 MPa.

The typical macroscopic plastic strain uncertainty seen

was measured to be 0.5 pct.

III. EXPERIMENTAL RESULTS

Experiments incorporating the X-ray technique

described in Section II were performed on both uniaxially and equibiaxially strained AA5182 specimens. First

we discuss the results for the uniaxial as-received and

20 pct balanced biaxially prestrained specimens, which

are used to calibrate the stress measurement technique.

Then, the balanced biaxial data are presented with and

without the XEC calibration for comparison. Finally,

we discuss some simple models often used to develop

biaxial behavior based on uniaxial behavior and compare these results with the biaxial data developed in the

present study.

A. Uniaxial

As described previously, a small tensile frame with a

load cell was built to test ASTM E8 subsized tensile

specimens in situ (under the X-ray beam) to determine

the ability of the X-ray system to measure stresses in

highly textured sheet samples undergoing plastic deformation. The true stress (r)–true strain (e) results for

AA5182 are shown in Figure 5 for the rolling direction

(RD, open circles) and transverse direction (TD, closed

circles) developed from the load cell and extensometer

data. Each data point shown is the result of one XRD

stress scan (recall that one scan includes multiple sample

angles and takes about 90 to 180 seconds). The data

shown are a result of multiple specimens with axial

loading oriented in each direction (rolling and transverse). A power law is fit to each data set (over the 2 to

15 pct strain range), and the fits are shown as solid

curves in Figure 5 and numerically in Table I. The

variability in the stress is not solely a result of

the different specimen tested, but is partially due to

the serrated yielding seen in AA5182.[19] Figure 6 is a

plot of the r-e trace for a typical ex-situ uniaxial

experiment (not under the X-ray system) with a constant

strain rate of 1 · 10)3 for a sample oriented transverse to

the rolling direction. The stresses measured by the load

Fig. 5—In-situ uniaxial results (circles) with power-law fit curves.

Table I. Power-Law (r = Ke

n

) Fit Parameters for Various Reported Data (MPa); Values Shown in Braces are Uncertainties

Based on One Standard Deviation of the Fit

Uniaxial (in-situ) Uniaxial (ex-situ) Biaxial (in-situ) Biaxial (in-situ with XEC Correction)

0 deg K 536 [3.53] 565 [3.44] 566 [9.52] 560 [8.35]

n 0.289 [0.0032] 0.302 [0.0015] 0.256 [0.0064] 0.272 [0.0059]

90 deg K 514 [2.59] 558 [4.26] 554 [6.85] 460 [9.04]

n 0.293 [0.0029] 0.316 [0.0016] 0.270 [0.0055] 0.198 [0.0072]

METALLURGICAL AND MATERIALS TRANSACTIONS A VOLUME 38A, FEBRUARY 2007—309
cell show clearly the serrated flow of AA5182, indicating

a negative strain rate sensitivity. The power-law fit

shown in Figure 6 is for the transverse orientation insitu uniaxial data (with multiple hold times, shown in

Figure 5). As expected, the data taken at the hold times

are recorded after initial relaxation, and thus lie on the

lower edge of the serrated flow curve. Note that the

values determined for the power-law exponent for both

orientations of in-situ data are nearly identical (Table I),

but that the stress values differ by about 5 pct.

Calibrated XECs are determined by plotting the slope

of the linear regression of the sin2

w data (P1) versus the

load cell measured true stress (r11 = P(DL/L)/Ao).

These data, shown in Figure 7, are linearly fit, and the

slope of the line equals the average effective XEC

(1 + m)/E. This calibration is needed to accurately

measure the stresses via XRD.[20] Note that ASTM

1426[20] describes the need for XEC determination.

However, the standard procedure subjects the calibration sample to elastic strains only, and uses a strain gage

and elastic modulus to measure the current stress for

calibration. Therefore, a modification of the standard

method is used here so that we can determine how the

XECs change with plastic strain level. The linear fit

shown in Figure 7 is determined using only the solid

points shown, which are along the hardening flow curve

(after initial yield to just before the localization prior to

failure). Table II lists the values of XEC taken from the

literature and the effective constants measured using our

system for both orientations (rolling and transverse) and

for 0 and 20 pct balanced biaxial prestraining. The

differences seen are attributed to the texture and

accumulated plastic strain effects on the XECs, which

are the subject of an ongoing study.

B. Balanced Biaxial

A series of tests were performed measuring in-situ

stresses at various imposed plastic strain levels during

balanced biaxial tension using the system default XECs.

These results are presented in Figure 8, and may be

contrasted to the flow data measured in uniaxial tension

Figures 5 and 6. In both the RD and TD, the flow stress

at a given biaxial strain level is higher than that

measured at the same uniaxial strain level. The coefficients to the power-law strain hardening relation were

calculated between 2 and 25 pct plastic strain and are

shown in Table I, and the resulting fits are plotted in

Figure 8 as a solid curve. The biaxial data was corrected

for the effective XECs measured previously (shown in

Table II) using linearly interpolated XEC values based

on the current plastic strain level. The corrected form of

Figure 8 is shown in Figure 9, where the average stress

levels are brought closer together but the hardening

indexes are seen to diverge (Table I). The yield stress

measured in this investigation (using the 0.2 pct strain

offset method) was compared to that measured by

Banabic[8] using cruciform testing and temperature

measurement to probe the initial yield locus, and these

values are also shown in Table III. It is seen that there is

a good correlation between the balanced biaxial tension

yield stresses reported in Reference 8 and the values

measured in the rolling and transverse directions in the

Fig. 6—Serrated yielding seen in uniaxial ex-situ results with a power-law fit of the in-situ results for specimens oriented transverse to

the rolling direction.

Fig. 7—Comparison of the slope (P1) of XRD measured lattice

strain vs sin2

w data and load cell measured stress for the in-situ uniaxial experiment in (a) the RD and (b) the TD. The slope of the linear fits shown here are the calibrated XECs, (1 + m)/E, for each

orientation. Only the solid points are used in the linear regression.

310—VOLUME 38A, FEBRUARY 2007 METALLURGICAL AND MATERIALS TRANSACTIONS A
present study. Note: the Banabic value assumes isostress

conditions.

C. Discussion

In simulations of metal forming operations, the

uniaxial strain hardening exponent from the power-law

hardening relation is often used to predict multiaxial

stresses. Multiaxial work hardening can be defined as a

change in effective stress with effective strain:

n 1⁄4 d ln ðreÞ

d ln ðee

pÞ 1⁄24

If one uses the von Mises’ criterion (assuming plane

stress, r3 = 0), the effective stress is given by

re 1⁄4

ffiffiffi

2

p

2 ð Þ r1  r2

2

þð Þ r2

2

þð Þ r1

2 h i1

2

1⁄25

in terms of the principal stresses (r1 and r2), and the

effective plastic strain is given by

e

e

p 1⁄4

ffiffiffi

2

p

3 e

p

1  e

p

2

2

þ e

p

2  e

p

3

2

þ e

p

3  e

p

1

2 h i1

2

1⁄26

in terms of the principal plastic strains ( e

p

1, e

p

2, and e

p

3).

For a uniaxial tensile test, r2 = 0 and e

p

2= e

p

3= –1/2 e

p

1

(assuming constant volume) . Note: the effective stressstrain curve is the same as the measured uniaxial curve

(re = r1 and ee

p 1⁄4 e

p

1) by design. For balanced biaxial

tension, e

p

1= e

p

2and r3 = 0, and assuming constant

volume, e

p

3= –2 e

p

1. Additionally, the normality requirement results in r1 = r2. Therefore, the effective values

are re = r1 and ee

p 1⁄4 2e

p

1. Because the effective values

are multiples of the uniaxial values, we can define a

multiplication factor (f) to convert the uniaxial response

to the predicted biaxial response:

fr 1⁄4 r1

ðreÞbiaxial

1⁄4 1 and fe 1⁄4 e

p

1

ðeeÞbiaxial

1⁄4 1

2 1⁄27

A similar procedure using the modified Hill criterion[21] would result in biaxial multiplication factors of

fr 1⁄4 ð2 þ 2rÞ

1=m

2 and fe 1⁄4 1

ð2 þ 2rÞ

1=m 1⁄28

Table II. X-Ray Elastic Constants (in GPa)1) from the Literature (for Various Aluminum Alloys) and Determined in the Present

Study (for AA5182) Considering Orientation and Balanced Biaxial Prestraining, ep;* Taken from Reference 11

Literature Calibrated Effective

RD TD RD TD

System Al 5083- Bulk ep = 0 ep = 0 ep = 20 ep = 20

Default H33* Al* Pct Pct Pct Pct

ð1þmÞ

E 0.01706 0.01817 0.02004 0.018294 0.015709 0.017628 0.018586

Fig. 8—Biaxial results using default XECs (in the RD and the TD)

with power-law fits shown as solid curves.

Fig. 9—XEC corrected biaxial results (in the RD, open circles, and

the TD, solid circles) with power-law fits shown as solid curves.

Table III. Comparison of Yield Strengths (in MPa) along Various Biaxial Strain Paths from the Present Study (by 0.2 Pct Offset

Method) and the Work of Banabic[8] (by Temperature Rise); the First Two Columns Compare Uniaxial Yield Points in the RD and

TD, the Second Two Columns the Plane Strain Results, and the Third Pair the Balanced Biaxial Results

r1 (RD) r2 (TD) PS (RD) PS (TD) BB (RD) BB (TD)

Banabic[8] 143 144 160 157 153 153

Present Study 133± 7 137 ± 7 — — 158 ± 12 163 ± 12

METALLURGICAL AND MATERIALS TRANSACTIONS A VOLUME 38A, FEBRUARY 2007—311
where r is the plastic strain ratio (measure of the

resistance to thinning, refer to ASTM E517[22]) and m is

an exponent deduced from experimental data. Banabic[8]

reports r-values for their AA5182 specimens, which are

used here to determine the weighted average r consistent

with ASTM E517, resulting in rm = 0.887. The original

Hill criterion had set m = 2 for all metals, but Parmar

and Mellor[21,23] report m values of 1.7 to 1.8 for soft

commercial aluminum. The resulting factors (f) may be

found in Table IV. Figure 10 is a plot of the biaxial

stress-strain data in the rolling direction (open circles)

with solid curves showing the in-situ uniaxial power-law

fit adjusted by the factors shown in Table IV and the

dashed curve is the power-law fit of the biaxial data

described in Table I. Using this effective stress and

strain construction to develop the biaxial mechanical

behavior curves does bring the uniaxial curves closer to

the biaxial data, but if one assumes power-law hardening r = Ke

n

, the effect of the multiplicative factors

(r1 = (fr/fe

n

) Ke1

n

) would result in a change in the K

parameter only. Therefore, the effective stress-strain

construction used with the power-law model is not

sufficient to predict the change in the strain hardening

exponents seen in the biaxial data and Table I. Similar

results may be shown for other common models

including the modified power-law model (r = k(e–eo)

n

)

and the Voce model ( r 1⁄4 S1ð1  Aee=ec Þ), where the

changes seen in the fitting parameters for the biaxial

data cannot be achieved through the effective stressstrain constructions described previously. The overall

result is an overprediction of the biaxial hardening by all

of the models considered. This overprediction is greater

at higher plastic strain levels, which are the strains

typically seen in forming operations.

The results of these analyses are not surprising if one

considers the assumptions of the models, which include

isotropic material behavior and isostress conditions in

balanced biaxial tension. The data in Figure 9 reveal

that neither of these results is seen experimentally. The

in-plane isostress condition required previously is not

met due to crystallographic texture effects, seen by

comparing the measured stresses in the rolling and

transverse directions in the balanced biaxial test

(Figure 9). Additionally, work hardening in balanced

biaxial tension is seen to be much lower than that

measured in uniaxial tension (Table I). These results are

not totally unique, as others have seen that the von

Mises’ criterion does not fit experimental data,[6] and

other laws such as Hill’s have been developed to account

for crystallographic texture via other factors such as the

r value.[22] However, none of these studies discuss the

possibility of a nonisostress state during balanced

biaxial straining. Data generated using the technique

developed in the present investigation are being used to

study how well current flow models predict the evolution

of the yield locus with increasing plastic strain while

admitting the possibility of mechanical behavior with

texture orientation dependence and nonequal principal

stresses during balanced biaxial straining.

IV. CONCLUSIONS

A novel methodology for measuring multiaxial,

in-plane stress-strain curves in sheet metal has been

presented. The method involved no modification of the

sheet beyond trimming to size and shape well outside of

the gage section and can be used on iron-, aluminum-,

and copper-based alloys. Balanced biaxial stress-strain

curves were measured in 5182 aluminum alloy sheet

samples, and the results are compared to the strengths

and hardening exponents of the same material measured

in uniaxial tension in the RD and TD of the sheet. This

method is currently being applied to determine how the

yield locus varies for a number of different alloys as a

function of plastic strain and multiaxial prestrains.

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Table IV. Biaxial Equivalent Multiplication Factors of

Uniaxial Data Based on Various Models for Model Parameters m and r Shown

von Mises’

Hill’s modified Hill’s

m = 2 and

r = 0.887

m = 1.75 and

r = 0.887

fr 1 0.971 0.967

fe 0.5 0.515 0.468

Fig. 10—Comparison of in-situ biaxial data in the rolling direction

(open circles and dashed curve) and model predictions using the

equivalent stress-strain construction and in-situ uniaxial power-law

fit in the rolling direction.

312—VOLUME 38A, FEBRUARY 2007 METALLURGICAL AND MATERIALS TRANSACTIONS A
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