Experimental observations of evolving yield loci

in biaxially strained AA5754-O

M.A. Iadicola *, T. Foecke, S.W. Banovic

National Institute of Standards and Technology, Gaithersburg, MD 20899-8553, USA

Received 12 July 2007; received in final revised form 29 February 2008

Available online 12 March 2008

Abstract

Experimental measurement of the plastic biaxial mechanical response for an aluminum alloy

(AA5754-O) sheet metal is presented. Traditional methods of multiaxial sheet metal testing

require the use of finite element analysis (FEA) or other assumptions of material response to

determine the multiaxial true stress versus true strain behavior of the as-received sheet material.

The method used here strives to produce less ambiguous measurements of data for a larger strain

range than previously possible, through a combination of the Marciniak flat bottom ram test and

an X-ray diffraction technique for stress measurement. The study is performed in conjunction

with a study of the microstructural changes that occur during deformation, and these microstructural results are briefly mentioned in this work. Issues of calibration and applicability are discussed, and results are presented for uniaxial (U), plane strain (PS), and balanced biaxial (BB)

extension. The results show repeatable behavior (within quantified uncertainties) for U to 20%,

PS to almost 15%, and BB to above 20% in-plane strains. The results are first compared with

three common yield locus models (von Mises’, Hill’48, and Hosford’79), and show some unexpected results in the shape change of the yield locus at high strain levels (>5% strain). These

changes include the rotation of the locus toward the von Mises surface and elongation in the

balanced biaxial direction. Comparison with a more complex yield locus model (Yld2000-2d with

eight adjustable parameters) showed that the locus elongation in the biaxial direction could be fit

well (for a specific level of work), but at the detriment of fit to the plane strain data. Artificially

large plastic strain ratios would be needed to match both the biaxial and plane strain behavior

even with this more complex model.

Published by Elsevier Ltd.

0749-6419/$ - see front matter Published by Elsevier Ltd.

doi:10.1016/j.ijplas.2008.03.003

* Corresponding author. Tel.: +1 301 975 5703.

E-mail address: mark.iadicola@nist.gov (M.A. Iadicola).

Available online at www.sciencedirect.com

International Journal of Plasticity 24 (2008) 2084–2101

www.elsevier.com/locate/ijplas
Keywords: C. Mechanical testing; B. Anisotropic materials; A. Microstructures; B. Polycrystalline material; B.

Metallic material

1. Introduction

Aluminum alloys have been receiving greater attention from the metal forming community in the last decade. AA5754 is of particular interest to the automotive industry for its

high ductility, light weight, and strength. Unfortunately, the use of this material presents

certain difficulties including surface roughening during deformation, occasionally yield

point phenomena (including an initial yield plateau), and the Portevin–Le Chatelier

(PLC) effect (Kang et al., 2006), which results in a serrated mechanical response during

plastic yielding. However, these difficulties are surmountable, and the use of any alloy

often is limited by its ability to be numerically modeled for an intended use. Many constitutive models exist based on extrapolation/prediction of multiaxial behavior from basic

uniaxial tension tests, but these predictions often are insufficient to predict the actual biaxial material behavior (Ofenheimer et al., 2007).

Some constitutive models are calibrated using data from multiaxial tension and/or

shear testing to achieve more accurate simulations (e.g., Vegter and van den Boogaard,

2006). However, the experimental data available are generally limited in both the strain

states measured and the level of strains achieved. For example, Vegter and van den Boogaard (2006) combine the data from multiple methods of testing, yet still have to make certain assumptions in regards to the expected mechanical behavior to develop a broad

enough data set for use in their model. Experiments using cylinders of material deformed

by combined tension, torsion, and internal pressure are useful to develop data for a wide

range of multiaxial strain states (Kuwabara et al., 2005), but would require the bending

and welding of the material, thus changing the material’s initial condition and behavior.

Biaxial tension data for sheet metal samples can be generated using cruciform specimens

(e.g., Kuwabara et al., 2002; Banabic, 2004), bulge testing (e.g., Gutscher et al., 2004), or

wide strip testing (which is capable of achieving strain states from plane strain to simple

shear, e.g., Flores et al., 2007), but each technique has aspects that limit its accuracy,

whether it is the assumption of a constitutive law when using finite element analyses to

interpret the results or uncertainties due to out of plane bending and inhomogeneous

strain fields.

The purpose of this study is to produce less ambiguous stress–strain measurements during large strain in-plane (biaxial) stretching of an as-received aluminum alloy (AA5754-O)

sheet metal sample. These results can then be compared with common model predictions

of yield loci (in stress space) at various plastic strain levels. A variation of the Marciniak

flat bottom ram test is used to impose near-linear strain paths. This method permits application of strain paths from uniaxial through plane strain to balanced biaxial (for plane

stress samples). Often this range of deformations requires multiple testing methods

(Kuwabara, 2007; Vegter and van den Boogaard, 2006), each with its own dissimilar

assumptions, thus leading to a potentially inconsistent data set. The method used here produces data that is measured in a self-consistent way. This method incorporates unquantifiable friction forces, thus there is no simple relation between the stress in the sheet and the

ram force. This problem is overcome by the use of X-ray diffraction (XRD) in a manner

similar to residual stress measurement to determine the in situ stress in the sheet.

M.A. Iadicola et al. / International Journal of Plasticity 24 (2008) 2084–2101 2085
2. Experimental technique

Below is a brief description of the material to be studied, and descriptions of the two

types of experiments to be performed: (1) standard tensile uniaxial fixed-rate tests and

(2) tensile multiaxial experiments with pauses in loading during which X-ray diffraction

measurements are made. The standard uniaxial tests are for comparison with the XRD

results, and are the typical/basic tests used to calibrate models for biaxial simulation in

the absence of multiaxial experimental data.

2.1. Material

The material used in this investigation is commercially available AA5754-O sheet. The

chemical composition determined using ASTM standard E227 is reported in Table 1. The

microstructure of the as-received sheet in the O-temper shows recrystallized grains, Fig. 1.

The size of the grains is relatively uniform in all three directions with those in the normal

plane having an average diameter of 39 lm±6 lm, measured by the linear intercept

method (ASTM E112, with one standard deviation of uncertainty). Their shape is relatively equiaxed in the rolling plane, but slightly elongated when viewed in the transverse

plane. The macrotexture of the as-received material is measured using an X-ray diffraction

technique on a conventional diffractometer with quarter Euler cradle. Cu Ka radiation

(wavelength k 1⁄4 0:1541 nm) is used to illuminate the samples with a multiwire area detector collecting the signal. The three-dimensional orientation distribution function (ODF),

Fig. 2, is expressed by Euler angles, /1; U, and /2 (in Bunge’s notation), which relate

Table 1

Chemical composition of AA5754, as mass fraction multiplied by 100

Si Fe Mg Mn Cu Zn Cr Al

0.06 0.24 3.56 0.28 0.05 0.05 0.05 Balance

Fig. 1. Optical micrographs (a) normal and (b) transverse to the rolling plane for the as-received material.

2086 M.A. Iadicola et al. / International Journal of Plasticity 24 (2008) 2084–2101
the cubic crystal frame to the orthotropic sample frame given by the normal direction

(ND), the rolling direction (RD), and the transverse direction (TD), respectively. Recrystallization components (Cube and R orientations), as well as a significant fraction of rolling texture components (Brass and Copper orientations), are observed.

2.2. Uniaxial experiments

Uniaxial (U) tests using ASTM E8 sub-size specimens (Fig. 3) are performed using a

88 kN electro-mechanical testing frame. Load during the testing is monitored using a calibrated 4.5 kN load cell. Strains are measured using two calibrated extensometers (one

Fig. 2. Orientation distribution function of as-received AA5754-O, with /1 and U axes running from 0to 90.

Positions of ideal orientation are shown (Cube, R, Goss, Brass, S, and Copper).

M.A. Iadicola et al. / International Journal of Plasticity 24 (2008) 2084–2101 2087
axial with a 10 mm gage length, and one transverse, across the width). Due to the PLC

banding seen in the material response and the resulting local inhomogeneous plastic strain,

control of the test is based on grip displacement, resulting in an average strain rate of

6 104 s1.

2.3. Biaxial experiments

The multiaxial tension experiments are performed using a variation (Raghavan, 1995)

of the Marciniak (Marciniak and Kuczyn ́ski, 1967) flat bottom ram test where each sheet

metal specimen is reinforced outside of the gage section with a sacrificial washer sheet (in

this case made of a mild steel). A modified 500 kN capacity servohydraulic mechanical

testing frame (Foecke et al., 2001) is used with specialized tooling (cylindrical ram and binder, Fig. 4) to perform the experiments. Balanced biaxial (BB) states of strain are achieved

with a 305 mm square specimen and washer (with a 32 mm diameter hole). Near-linear

strain paths for other strain states are realized using a similar geometry, but with a reduced

specimen width along the direction of reduced straining (Raghavan, 1995). For example,

near plane strain (PS) conditions are achieved here using a specimen width of 135 mm

(with a longitudinal length of 305 mm), resulting in strain in the longitudinal direction

while having near zero strain in the transverse direction (width). Due to the hole in the

washer and a recess in the center of the ram (Fig. 4), the gage section (center) of the specimen is stress free on the upper and lower surfaces. Strain is measured using a calibrated

biaxial extensometer (Class C, ASTM E83-92, accuracy throughout the 40% strain extensometer range).

6.35

25.4

12.7

>100

R 6.35

1.0

Fig. 3. Uniaxial specimen with dimensions shown in mm.

66o

Specimen

Washer

100

12

12

X3

X1

X2

X1

Lower

Binder

Upper

Binder

Ram

Binder

Ram

A A

Top View A-A

Lock Bead

152.4

Lock Bead 110

Fig. 4. Marciniack tooling (dimensions shown in mm) with specimen inserted in cross-section view (right).

2088 M.A. Iadicola et al. / International Journal of Plasticity 24 (2008) 2084–2101
As mentioned earlier, due to the presence of unquantifiable friction forces, no simple

relation between ram force and applied stresses exists. Therefore, a method of stress determination (often referred to as the ‘‘sin2 w method”) using XRD methods and equipment

originally developed for residual stress measurement is used. This method (described in

Foecke et al., 2007), uses an X-ray beam source (typically Co Ka or Cr Ka) diffracted from

a sampled volume on the surface of the specimen to measure the interatomic spacing

(through Bragg’s Law) from a set of atomic planes at a specific orientation. This method

samples only the crystallites in the diffracted volume with a specific selected crystallographic orientation with respect to the beam and detector (Fig. 5). By tilting the beam

and detector an angle ðwÞ to the surface normal, a different set of crystallites within the

diffracting volume will be sampled. By oscillating the w angle slightly ð3Þ, the number

of crystallites sampled are increased within the neighborhood of the ideal orientation of

interest. This results in a peak profile observed using two-256 channel, 10position sensitive scintillation detectors (one placed symmetrically on each side of the source beam). The

individual detector channels are calibrated using a scan of an amorphous target (a glass

slide in this case). After detector gain correction, a background level is determined based

on two points selected outside of the peak limits. In the current work, the upper 85% of the

peak is fit to a Pearson’s VII function (with an exponent of 1.77), after the background

level of the profile is subtracted. Peak broadening (due to an increase in dislocation density

during plastic deformation) may not allow for proper background determination. Care is

taken to ensure that for each measurement the peak width is never more than half of the

detector width, and that the tails of the peak are not cropped by the limits of the detector.

For all the in situ XRD data shown here, the source radiation is the Co Ka (wavelength

k 1⁄4 0:1790 nm), and the reflection is for the {420} family of planes (Bragg angle

2h 1⁄4 162:5).

The individual crystallites sampled respond elastically to the applied load even during

dislocation motion, and after the specimen is yielded through the thickness, the surface

X3

X3

X1

X1

^

^

ψ

Detector arc

Collimator

Detector

X-ray Beam

Specimen

Surface

X3

φ

ψ

φ

^

X3

X1

X2

^

X2

^

X1

ψ

Specimen

Surface

Fig. 5. XRD axes definition, where (a) is the general definition and (b) is for / 1⁄4 0 and shows the X-ray beam,

collimator, and detector locations. The X axes define the specimen system, and the Xb axes define the detector

system.

M.A. Iadicola et al. / International Journal of Plasticity 24 (2008) 2084–2101 2089
stress equals the macroscopic flow stress through the thickness. The diffracting volume

must be chosen sufficiently large to incorporate a representative sample of crystallites in

various crystallographic orientations. Here the focal spot illuminated is 5 mm long and

1.5 mm wide (in the w tilt direction) with a maximum penetration depth of approximately

40lm. This results in a diffracting volume encompassing approximately 4500 grains (based

on the average grain size). The intensity variation with position on the surface of an asreceived sample may be used to approximate the number of crystallites sampled (Gnaeupel-Herold et al., 2004) for a given w tilt angle. A limited data set of these intensities for

our material results in a conservative sampling estimate of greater than 150 grains for

almost all of w tilt angles used, and never less than 70 grains. Although the data set is

too limited to achieve statistically significance, even the conservative estimate is sufficiently

large to be a representative sampling, and this adequacy is reinforced by the quality of the

diffraction peaks measured throughout this study.

The standard governing equation relating the lattice strain to the surface strains (Noyan

and Cohen, 1987) in the surface system (Fig. 5a) where the X1 axis parallel to the rolling

direction of the sheet is

d/w  do

do

1⁄4 e11 cos2 /sin2 w þ e12 cos 2/ sin2 w þ e22 sin2 /sin2 w þ e33 cos2 w

þ e13 cos / sin 2w þ e23 sin /sin 2w; ð1Þ

where d is the interatomic lattice spacing determined through Bragg’s Law, / (the rotation

about X3) and w (the rotation about Xb 2) are the orientation angles, and eij are the components of strain in the surface system. The stress-free lattice spacing ðdoÞ is not measured

directly, since the ‘‘sin2 w method” is not very sensitive to error in this value, and a system

default value for do of aluminum is assumed. To verify this, an XRD measurement of a

stress-free aluminum powder is made, and the assumed do is considered acceptable if

the measured stress is less than the uncertainty of the measurement.

Since the top and bottom surfaces of the specimen are stress free in the center region of

interest, we neglect the through thickness stress ðr33 1⁄4 0Þ. Assuming an isotropic constitutive law and tilting in the rolling direction ð/ 1⁄4 0Þ Eq. (1) becomes

dw  do

do

1⁄4 S2r11 sin2 w þ S1ðr11 þ r22Þ þ S2r13 sin 2w ð2Þ

and if the tilt is in the transverse direction ð/ 1⁄4 90Þ

dw  do

do

1⁄4 S2r22 sin2 w þ S1ðr22 þ r11Þ þ S2r23 sin 2w; ð3Þ

where S1 1⁄4  m

E and S2 1⁄4 1þmE . S1 and S2 are the effective X-ray elastic constants (XECs)

whose values are associated with a chosen family of planes ðfhklgÞ in a specific material,

and may be affected by alloying or crystallographic texture (Noyan and Cohen, 1987), as

well as by other less significant variables. It is reasonable to assume that the effective XECs

will differ for the rolling and transverse directions; therefore we experimentally calibrate

for S2 in each of these directions. We are most concerned with the stress in the direction

of tilt ðwÞ and will perform multiple tests to obtain the stress in each of the first two principal strain directions, which are chosen here to coincide with the rolling or transverse

directions of the sheet material. These measurements will require calculation of the slope

of the lattice strain – sin2 w relationship, but not the intercept, thus S1 does not need to be

2090 M.A. Iadicola et al. / International Journal of Plasticity 24 (2008) 2084–2101
determined. (Note: although the second terms in Eqs. (2) and (3) are not measured, they

are used by the XRD system to permit the combination of the data from the two detectors

by assuming a common intercept in the lattice strain – sin2 w behavior.) Since the top and

bottom surfaces of the sample are stress free and the measurements are made after the initial yielding of the material, no shear stresses r13 or r23 should exist, and any lattice strain

variation with sin 2w is a result of systematic errors or micro-stresses. In our results, these

effects are minor (within the uncertainty of the XRD measurements), and therefore are not

discussed further.

The central procedure for the Marciniak-style testing performed here includes three

steps: (1) the material is loaded by an increase in ram height at a quasistatic rate, (2)

the ram position is held and the XRD system is focused on the surface of the sheet,

and (3) a scan of selected w angles (13 in this case) is taken each with a small ð3Þ oscillation and an associated peak location (h angle) may be calculated (as well as the lattice

spacing through Bragg’s Law). At this point the stress may be calculated from the

XRD data. The macroscopic strain during this hold time is calculated by averaging the

output of the biaxial extensometer for the entire scan time, and typically has a standard

deviation <0.005% strain (which is 10 times larger than the typical fixed error associated

with the extensometer). This procedure can then be repeated for the next data point. The

entire procedure (1 through 3) takes approximately 8 min with each peak profile taking

about 15 s. This is due to the low power of the X-ray tube (60 W) and X-ray absorption

by the aluminum. Note: the XRD method of stress measurement described above is generally applicable, and although it is used here in conjunction with the Marciniak method

of imposing strains on an aluminum alloy, it could also be applied to other materials and

methods of biaxial straining (e.g., cruciform testing) as long as the material has been

yielded through the thickness without substantial gradients in stress through the thickness.

Effective XEC calibration experiments are performed using the uniaxial ASTM E8 subsize specimens (Fig. 3) prepared in the same manner as the specimens mentioned above. A

portable mechanical testing frame is used to strain the samples while under the XRD system. Load (measured using a calibrated load cell) and strain (measured using an axial

extensometer) are used to determine the true stress in the gage section. At selected strain

levels the specimen is unloaded (to a level never less than 80% of the initial uniaxial yield)

and reloaded elastically. During these elastic unload/reload times, the uniaxial loading is

paused at multiple points and XRD is used to measure the current stress. By comparing

the XRD and load cell measured values, the calibrated effective XECs may be determined.

For this work, the average effective XECs are determined by averaging all of the XECs

measured for each elastic unload/reload (7–8) during one uniaxial test in each direction

(RD and TD). Since texture evolves during plastic deformation (Banovic and Foecke,

2003; Banovic et al., 2008) and effective XECs may be texture dependent, it would be

expected that these values vary with strain or stress level and/or path. This has been shown

to be a minor effect in this case (Iadicola and Gnaeupel-Herold, 2008) and a detailed discussion is beyond the scope of this work.

3. Results

Stress–strain curves for the standard tensile uniaxial fixed rate tests are shown in Fig. 6

for the rolling (RD), transverse (TD), and 45diagonal (DD) directions. Three tests are

performed for each direction, and all result in nearly identical curves. Therefore, only

M.A. Iadicola et al. / International Journal of Plasticity 24 (2008) 2084–2101 2091
one representative curve is shown in Fig. 6 for each orientation. These data are fit with

power law ðr 1⁄4 KenÞ and Voce ðr 1⁄4 S1ð1  AeBe

ÞÞ hardening models over the applicable

plastic range (above the initial yield and below the maximum uniform elongation), and are

shown as dashed and dotted lines, respectively in Fig. 6 (Table 2 lists the average values of

these model variables, calculated from the three repeat tests). The Voce fits for all three

directions are plotted together in Fig. 6d to better show the relative position of the data

plotted in Fig. 6a–c. Since PLC banding occurs, the elongation is only truly uniform in

the elastic region, but we assume a definition of the maximum uniform elongation based

on the maximum point of the engineering load displacement curve. A further adjustment is

made in selecting the maximum point of the engineering load displacement curve. Since

the serrated yielding (seen throughout the plastic yield regime) tends to cause sudden local

maxima and local minima, the actual maximum load recorded may be the effect of local

0 5 10 15 20 25 30

 ε (%)

0

50

100

150

200

250

300

(MPa)

σ

 continuous

 Power Law Fit

 Voce Fit

 XRD

average maximum load 

0 5 10 15 20 25 30

 ε (%)

0

50

100

150

200

250

300

(MPa)

σ

 continuous

 Power Law Fit

 Voce Fit

 XRD

average maximum load 

0 5 10 15 20 25 30

 ε (%)

0

50

100

150

200

250

300

(MPa)

σ

 continuous

 Power Law Fit

 Voce Fit

average maximum load 

0 5 10 15 20 25 30

 ε (%)

0

50

100

150

200

250

300

(MPa)

σ

 RD Voce Fit

 TD Voce Fit

 DD Voce Fit

Fig. 6. Uniaxial true stress–true strain curves for specimen (a) 0, (b) 90, and (c) 45to the rolling direction, and

(d) the Voce fits for all three orientations plotted together. Data points (circles) are the result of uniaxial

experiments incorporating the XRD system.

2092 M.A. Iadicola et al. / International Journal of Plasticity 24 (2008) 2084–2101
PLC banding and not the overall response of the material. Therefore, the maximum uniform elongation and stress at that point are determined using the median values of true

stress and true strain within the range of data points where load is within 22 N (approximately equal to twice a typical load drop due PLC motion) of the actual maximum load.

This method greatly reduces the scatter between the three repeat tests in the variables

shown in Table 2.

The contraction ratio (a plastic analog to the Poisson’s ratio in elasticity, defined as the

negative of the width strain, ew, over the longitudinal strain, el) is reported in Table 2

where the value is averaged for the three experiments performed in each orientation over

the applicable plastic range (described above). The average value of r (the plastic strain

ratio, equal to the width strain, ew, over the thickness strain, et) as defined in ASTM

E517-92 is similarly calculated, where et is calculated from the measured strains assuming

constant volume during plastic deformation. The uncertainties reported for these ratios

are the standard deviations between the three experimental results for each orientation,

and not the variation seen in one experiment. The averaging for each experiment individually is required, since the transverse extensometer monitors the width of the specimen

over a much smaller amount of the axial length than the axial extensometer, resulting

in oscillations throughout the plastic range of the q and r values (as shown in Fig. 7) resulting from the PLC banding.

Additional uniaxial experiments are performed using a portable mechanical testing

frame with the XRD system. These are used to compare with the standard uniaxial curves

and to check the calibrated effective XECs ðSRD

2 1⁄4 19:3  0:9 TPa1

;

STD

2 1⁄4 19:7  0:7 TPa1

Þ. The data are plotted as points along with the standard uniaxial

experiments in Fig. 6a and b. The data match well with only a slight deviation between 5%

and 12% strain for both the RD and TD. This may be in part due to the effect of texture

changes during plastic deformation, and is discussed in Iadicola and Gnaeupel-Herold

(2008). Although the texture change itself was shown to have a minor effect, the subsequent intergranular stresses that arose were measurable, but close to the resolution of

the XRD system. The resulting variation in the XECs was deemed minor in comparison

to the uncertainty of the measurement (Iadicola and Gnaeupel-Herold, 2008) and therefore is neglected here. The error bars shown in Fig. 6a and b are based upon the quality

Table 2

Basic uniaxial material properties

Units RD TD DD

0.2% Yield stress MPa 94.1 [3.3] 92.1 [0.7] 90.9 [0.6]

Tensile strength MPa 226 [3] 218 [1] 216 [3]

Maximum uniform elongation % 18.9 [0.9] 21.5 [0.5] 22.4 [0.9]

q 0.409 [0.005] 0.464 [0.005] 0.422 [0.010]

r 0.69 [0.02] 0.87 [0.02] 0.73 [0.03]

Power law fita K MPa 474 [9] 458 [6] 447 [3]

n 0.317 [0.010] 0.326 [0.006] 0.323 [0.004]

Voce fitb S1 MPa 289 [2] 284 [2] 286 [3]

A 0.686 [0.014] 0.695 [0.007] 0.693 [0.004]

B 12.3 [0.3] 11.2 [0.2] 10.2 [0.5]

Uncertainties (shown in brackets) are based on one standard deviation.

a r 1⁄4 Ken. b r 1⁄4 S1ð1  AeBe

Þ.

M.A. Iadicola et al. / International Journal of Plasticity 24 (2008) 2084–2101 2093
of the linear fit of the lattice strains versus sin2 w. The size of the error bars are not much

larger than the stress variations resulting from the serrated yielding. The XRD data values

are predominantly at the lower edge of the serrated hardening curve, which seems intuitive

considering they are measured during pauses in loading. The load rises and drops are associated with PLC band creation and motion (both transient events). The same uniaxial data

are plotted in Fig. 8a along with the data for the PS and BB tests shown in gray for comparison. The points are connected by lines to aid visibility, but these lines are not intended

to suggest the actual stress strain path between the points.

The results for the plane strain (PS) experiments using XRD are plotted in Fig. 8b,

for both maximum strain in the rolling direction (PS in RD) and the transverse direction (PS in TD). The PS stress data for the near zero strain direction (with measured

strain ratios typically <0.05) are plotted using the strain in the maximum (principal)

strain direction for visibility, and are the sets of points below 100 MPa. The XECs

used are the same as those for the uniaxial experiments above. Since the XRD method

only measures the stress in one direction for each experiment performed, two experiments are needed to obtain (1) the stress in the direction of principal strain and (2)

the stress in the other (near zero strain) direction within the plane of the sheet. Two

experiments (one original and one repeat) are performed for each direction (the maximum strain direction and the near zero strain direction) for a total of eight experiments shown in Fig. 8b. Repeat experiments are performed by a different system

operator four months after the first set in order to verify the repeatability of the experimental method. There is a slight difference between the results of the repeat experiments, but most are within the error bars of the data. The trends are quite

consistent; for example the stress in the near zero strain direction is always higher

in the ‘‘PS in TD” experiments than in the ‘‘PS in RD” experiments.

0 5 10 15 20

 ε (%)

0.3

0.35

0.4

0.45

0.5

0.55

q 

RD

TD

DD

0 5 10 15 20

 ε (%)

0.5

0.6

0.7

0.8

0.9

1.0

 r 

RD

TD

DD

Fig. 7. Plastic strain ratios (a) q and (b) r variation with true strain (in applicable plastic region) for specimen 0

(RD, solid line), 90(TD, dashed line), and (c) 45(DD, dotted line) to the rolling direction. Results are for the

same experiments as shown in Fig. 6.

2094 M.A. Iadicola et al. / International Journal of Plasticity 24 (2008) 2084–2101
The balanced biaxial experimental results are plotted in Fig. 8c for the sheet texture

directions (RD and TD). Again a separate experiment is performed for each direction

and repeat experiments are performed similarly to those above, all using the same uniaxially determined XECs. The repeat experiments show good agreement with the exception

of one data point in the RD at about 2.7% strain, which appears to be about 15 MPa too

low. The general trend is that the stress in the TD is initially higher than the RD up to 5%

strain at which point the trend reverses and the RD stress is higher.

As mentioned earlier, the ‘‘sin2 w method” for stress determination can be sensitive to the

presence of crystallographic texture, as texture will have a tendency to increase (or decrease)

the number of grains oriented at the Bragg condition for a given tilt, thereby increasing (or

decreasing) the diffracted X-ray signal. Additionally in some materials, texture has been

shown to cause the unusual nonlinearity in the lattice strain – sin2 wrelationship (e.g., for iron

or steel Do ̈lle et al., 1977; Marion and Cohen, 1974). Additional samples of AA5754-O are

0 5 10 15 20

 ε (%)

0

100

200

300

400

(MPa)

σ

BB in RD

BB in TD

PS in RD

PS in RD

PS in TD

PS in TD

 U in RD

 U in TD

0 5 10 15 20

 ε (%)

0

100

200

300

400

(MPa)

σ

BB in RD

BB in TD

U in RD

U in TD

 PS in RD

 PS in TD

0 5 10 15 20

 ε (%)

0

100

200

300

400

(MPa)

σ PS in RD

PS in RD

PS in TD

PS in TD

U in RD

U in TD

 BB in RD

 BB in TD

Fig. 8. Multiaxial true stress–true strain curves determined through XRD for (a) U, (b) PS, and (c) BB

experiments with other experiments shown in gray for comparison. PS stress data for the near zero strain

direction are plotted using the first principal strain for visibility.

M.A. Iadicola et al. / International Journal of Plasticity 24 (2008) 2084–2101 2095
prestrained (using the same Marciniak tooling) to intermediate strain levels (5%, 10%, 15%,

and 20%) for microscopy and macrotexture analysis. Fig. 9 shows a cross-section of the

AA5754-O sheet strained to 20% in-plane plastic strain. The elongation of the grains, due

to plastic deformation, is obvious if compared to Fig. 1b. Fig. 10 shows three /2-sections

for the as-received material and those resulting after biaxial extension (to 5%, 10%, 15%,

and 20% strain). For a complete review of the changes seen in texture and microscopy for

the material in this study, see Banovic et al. (2008). In Fig. 10 some orientations are noticeably strengthened (components in the h011iND-fiber, 0 6 /1 6 90; U 1⁄4 45, and /2 1⁄4 90)

and others weakened (recrystallization components, R, Cube). Despite these distinct

changes in texture (e.g., the decrease in Cube texture is greater than four times the random

distribution) and grain shape, the method does not show any catastrophic loss of signal or

non-linear sin2 w behavior for this alloy.

4. Discussion

Fig. 11 is a plot of the multiaxial data interpolated into levels of constant plastic work

on stress axes. The data points are developed using the data in Fig. 8 and the definition of

plastic work as

W 1⁄4

Z

rij  1

3

rkkdij d e

p

ij  1

3

e

p

kkdij ; ð4Þ

where the first term on the right hand side is the deviatoric stress, the second term in the

deviatoric plastic strain, and the indices range from 1 to 3 (with repeated indices indicating

summation). Assuming constant volume during plastic straining, this may be simplified to

W 1⁄4

Z

rij  1

3

rkkdij de

p

ij; ð5Þ

where plastic strain ðepÞ is calculated using the measured true strain and true stress

e

p

ij 1⁄4 eij  1þm

E rij þ mE rkkdij   assuming elastic constants E 1⁄4 70 GPa and m 1⁄4 0:33. Since

Fig. 9. Optical micrograph of 20% balanced biaxial strained sample transverse to the rolling plane.

2096 M.A. Iadicola et al. / International Journal of Plasticity 24 (2008) 2084–2101
Fig. 10. Selected ODF /2-sections for as-received material and after biaxial straining to four strain levels (5%,

10%, 15%, and 20%), with /1 and U axes running from 0to 90.

0 100 200 300

 σRD (MPa)

0

100

200

300

(MPa)

σ TD

PS

BB

PS

ε = 1%

5%

10%

15% - von Mises’

 ε = 1%

 5%

 10%

 15%

0 100 200 300

 σRD (MPa)

0

100

200

300

(MPa)

σ TD

PS

BB

PS

ε = 1%

5%

10%

15% - Hill’48

ε = 1%

5%

10%

15% - Hosford’79

 ε = 1%

 5%

 10%

 15%

Fig. 11. Data from Fig. 8 interpolated onto stress axes (data points) for comparison to (a) von Mises’ (solid

lines), (b) Hill’48 (solid lines) and Hosford’79 (dotted lines) model yield surfaces based on the uniaxial

experimental results. Dashed lines show the isotropic prediction for PS and BB strains.

M.A. Iadicola et al. / International Journal of Plasticity 24 (2008) 2084–2101 2097
the data in Fig. 8 is not continuous (and from as many as 4 different tests for one strain

state), interpolated curves for each data set (with some minor smoothing) are used to calculate the resulting plastic work. The values of plastic work per unit volume calculated for

the uniaxial RD at 1%, 5%, 10%, and 15% strain levels are used as the work contour values

of interest (with values of 0:69 MJ=m3; 6:43 MJ=m3; 16:52 MJ=m3, and 28:60 MJ=m3,

respectively). For these strain levels, the yield surfaces are calculated and plotted for three

models: von Mises’ (solid line in Fig. 11a), Hill’48 (Hosford and Caddell, 1983, pp. 266 ff.,

solid line in Fig. 11b), and Hosford’79 (Hosford and Caddell, 1983, pp. 269 ff., dotted line

in Fig. 11b, where the model parameter a 1⁄4 8 is used for an FCC material) using the plastic strain ratios, r values, from Table 2 when required by the models. The yield locus data

points plotted are those calculated from the data shown in Fig. 8 for the same plastic work

values of interest. The dashed lines in Fig. 11 show the location of the predicted PS and BB

data based on isotropic material behavior.

The XRD data for PS in the TD and RD are consistently offset from the isotropic prediction toward the uniaxial stress axes, where the offset is greater in the case of PS in the

RD. However, both sets of data lie on or close to the von Mises’ and Hill’48 curves. For

PS in the TD, Hill’48 (which incorporates the anisotropy using the r values in Table 2) is

closer to the data then von Mises’, but the Hill’48 does not match the data for PS in the

RD or U in the TD as well as the von Mises’ model. The BB XRD data matches the isotropic prediction well (generally following the rRD 1⁄4 rTD dashed line) with some minor

variation, but does not match any of the model curves in value, especially for the 10%

and 15% points (biaxial strains where data has typically been previously unavaliable for

sheet metal). There is some movement of the points from the rRD < rTD to rRD 1⁄4 rTD with

increased plastic work. The data suggests that the shape of the yield locus does not remain

constant with increasing plastic strain level, and thus does not match any of the models

(whether isotropic or anisotropic). The initial yield locus is flattened slightly in the balanced biaxial direction, and more convex toward the transverse stress direction near the

measured plane strain data. As plastic straining proceeds, the shape of the locus elongates

in the balanced biaxial direction substantially, while the other points approach or remain

close to the von Mises’ curves and to a lesser extent the Hill’48 curves. None of the models

match the BB stresses above the 1% strain level. Note: similar BB behavior is seen in steels,

but steel r values are >1 unlike aluminum alloys (values in Table 2 are more typical). Artificially large r values could be used to adjust the model predictions closer to the yield locus

data, but the r values would have no physical significance.

These yield locus changes are more clearly seen when plotting the differential hardening

where the axes are normalized by the uniaxial RD stresses at the same plastic work levels

r1⁄4 r

rRDðrTD1⁄40Þ

ð6Þ

as shown in Fig. 12. Where as the model curves each collapse to a single curve, the yield

points for each plastic work level (1%, 5%, 10%, and 15% strain) tend to shift closer to the

von Mises’ curve for both PS directions and for U in the TD, but the BB points rotate

toward rRD 1⁄4 rTD while extending further out from the model curves. This shows that

for these strain states the material behaves less anisotropically at higher strain levels (moving from orthotropic behavior to an almost specially orthotropic response). In all cases,

the major shift is seen between the 1% and 5% levels and almost no change is present

between the points associated with the 10% and 15% levels. In Kuwabara et al. (2005),

2098 M.A. Iadicola et al. / International Journal of Plasticity 24 (2008) 2084–2101
a different 5000-series aluminum alloy (initially processed into tubes) is deformed by combined pressurization and tension, and the data showed BB strain resulted in a rRD to rTD

ratio of 20:23 (0.87) for all the plastic strain levels from 0.2% to 25%. The results in Figs.

11 and 12 have rRD to rTD ratios of 0.85, 0.89, 0.98, 1.03 for 1%, 5%, 10%, and 15% strain,

respectively, and when including the intermediate strain levels (not plotted) the largest

changes are seen between 6% and 8% strain after which the ratios fluctuate in a range

of 0.98 to 1.03 (where rRD 1⁄4 rTD is within the uncertainties of the data points). Additionally, the results described in Kuwabara et al. (2005) do not show rRD > 1 and rTD > 1 as

seen here. It is not clear if these differences are a result of the different material tested (alloy

and heat treatment) or the test configuration (sheet/tube) and its required assumptions.

A model that admits normalized BB stresses >1 is Yld2000-2d (Barlat et al., 2003). This

model is typically calibrated (as described in Barlat et al., 2003) using standard uniaxial

tests (in the RD, TD, and DD) and a balanced biaxial stress test (where the strain ratio

of two directions is defined as rb 1⁄4 deTD=deRD). An average rb value of 1.13 is determined

from the data in Fig. 8c. The stress values for the model calibration are taken from the

data in Fig. 12 with the exception of the uniaxial DD values which are determined from

the data in Fig. 6c with the stress values reduced by 6.5 MPa to center the data on the

lower edge of the PLC load drops similar to the XRD data (shown in Fig. 6a and b

and discussed in Section 3) such that the uniaxial DD curve is very close to or below

the uniaxial XRD TD data shown in Fig. 8a similar to the relative curve (dotted and

dashed) locations shown in Fig. 6d. The resulting model parameters (using a 1⁄4 8 for

FCC material, as before) calculated for the 15% data in Fig. 12 are

a1 1⁄4 0:8319; a2 1⁄4 1:0765; a3 1⁄4 0:6731; a4 1⁄4 0:9592;

a5 1⁄4 0:9530; a6 1⁄4 0:6520; a7 1⁄4 0:9784; and a8 1⁄4 1:4059

and the associated yield locus curve is plotted in Fig. 13b (Case 1, solid line). Because the

model calibration uses the BB and U data these data are matched well. The model predictions for PS, however, are far off of the experimental data. Using artificially high values of

rRD 1⁄4 1:5 and rTD 1⁄4 3:0, while holding the other measured values fixed, a closer fit to the

experimental data results, but rRD and rTD now have no physical significance (as mentioned earlier). This change results in the revised model parameters:

0 0.5 1.0 1.5

 σRD

 _

0

0.5

1.0

1.5

σ TD

_ 

von Mises

Hill’48

Hosford’79

 ε = 1%

 5%

 10%

 15%

Fig. 12. Data from Fig. 11 normalized by uniaxial yield stresses in the rolling direction with the von Mises’ (solid

curve), Hill’48 (dashed curve), and Hosford’79 (dotted curve) predictions. The dashed line traces equal stress in

RD and TD, for reference.

M.A. Iadicola et al. / International Journal of Plasticity 24 (2008) 2084–2101 2099
a1 1⁄4 0:8837; a2 1⁄4 1:1993; a3 1⁄4 0:8542; a4 1⁄4 0:8897;

a5 1⁄4 0:9227; a6 1⁄4 0:6599; a7 1⁄4 0:9775; and a8 1⁄4 1:4077

and the second yield locus curve plotted in Fig. 13b (Case 2, dotted line). Fig. 13a shows

these same cases plotted at the all work levels on true stress axes (similar to Fig. 11). These

predictions do not fit the PS values for the measured r values (Case 1), but are much closer

for the artificial r values. However, neither Case 1 nor Case 2 capture the behavior for

<15% strain. It would seem that the model parameters would have to change with strain

hardening to match all the work levels shown.

5. Conclusion

Experimental results for multiple tensile strain states (uniaxial, plane strain, and balanced biaxial) using X-ray diffraction as the measure of stress are shown. The uniaxial

data compares well with typical uniaxial tests, verifying the measured effective X-ray elastic constants used. Repeat tests show consistent results in measured values and trends for

all the tests performed. The yield surfaces show similar trends to those predicted by a basic

von Mises’ model, but some behaviors are unique, especially in the balanced biaxial direction. Comparison to the Hill’48 and Hosford’79 models showed little improvement over

the von Mises’ model. The Yld2000-2d model (with eight parameters) is able to model

the balanced biaxial results for a given level of work, but under predicts the plane strain

data for the measured plastic strain ratio, r, values. Future work is planned to include a

detailed investigation of the effect of texture changes on the mechanical behavior using

polycrystalline modeling.

Acknowledgements

The authors would like to thank T. Hamblin and T. Warren (Alcan International Ltd.)

for supplying the material used in this study, and F. Barlat (Pohang University of Science

and Technology) for calculation of the Yld2000-2d model parameters.

0 100 200 300

 σRD (MPa)

0

100

200

300

(MPa)

σ TD

PS

BB

PS

ε = 1%

5%

10%

15% - Case 1

1%

5%

10%

15% - Case 2

 ε = 1%

 5%

 10%

 15%

0 0.5 1.0 1.5

 σRD

 _

0

0.5

1.0

1.5

σ TD

_ 

Case 1

Case 2

 ε = 1%

 5%

 10%

 15%

Fig. 13. Equal plastic work yield stress data from Fig. 11a, and normalized values from Fig. 12b, both with

Yld2000-2d predictions for Case 1, measured r values (solid curves) and Case 2, artificially large r values (dotted

curves).

2100 M.A. Iadicola et al. / International Journal of Plasticity 24 (2008) 2084–2101
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