Constitutive Relations for AA 5754 Based on Crystal Plasticity

L. HU, A.D. ROLLETT, M. IADICOLA, T. FOECKE, and S. BANOVIC

Constitutive equations for the multiaxial stress-strain behavior of aluminum alloy 5754 sheets

were developed, based on crystal plasticity. A Taylor-based polycrystal plasticity model, a

tangent formulation of the self-consistent viscoplastic model (VPSC), and an N-site viscoplastic

model based on the fast Fourier transform (VPFFT) were used to fit a single slip system

hardening law to the available data for tension, plane strain, and biaxial stretching. The fitting

procedure yields good agreement with the monotonic stress-strain data, with similar parameter

values for each model. When simulating multiaxial tests using the developed hardening law,

models that allow both stress and strain variations in grains give better predictions of the stressstrain curves. Furthermore, generally, the simulated texture evolution is too rapid when compared to the experiments. By incorporating a more detailed neighbor interaction effect, the

VPFFT model predicts texture evolution in better agreement with experiments.

DOI: 10.1007/s11661-011-0927-1

The Minerals, Metals & Materials Society and ASM International 2011

I. INTRODUCTION

THE 5000-series magnesium-strengthened aluminum

alloys are used for interior sheet components in automotive bodies. They have excellent weldability and

corrosion resistance, providing a higher strengthto-weight alternative to steel that can enhance fuel

economy, safety, and driving performance.[1] Simulations of forming processes are needed to manufacture

parts efficiently from sheet aluminum metal. Such

processes often involve complex loading paths. Therefore, accurate predictions of stress-strain relations and

texture development, especially for multiaxial deformation and different strain paths, are required for reliable

simulations of metal forming processes. In addition,

knowledge of how the texture evolves with multiaxial

plastic strain, and having a robust mathematical

description of this behavior, is also useful as an input

to thermomechanical processing of the sheet as a control

of initial texture to improve formability.

Polycrystal plasticity models assume that strain hardening occurs through dislocation accumulation on each

slip system with varying degrees of complexity of

interactions among the systems in each grain. Thus,

the basic constitutive law adopted for this work is that

which describes the hardening behavior of an individual

slip system.

Taylor-type models assume identical strains across the

polycrystal.[2]

Asaro and Needleman[3]developed a ratedependent constitutive model based on the isostrain

Taylor assumption that could account for arbitrary

large strain and individual grain rotation. With a code

similar to the Taylor model code used in this work,[4]

Kocks et al.

[5] showed that polycrystal plasticity could

be used to account for the differences in hardening

between different strain modes in copper. Bronkhorst

et al.

[6]

made predictions of pole figures and stress-strain

curves, which they compared to copper experimental

data from compression, tension, simple shear, and plane

strain compression (only the stress and strain in the

longitudinal direction considered) tests. It was shown

that the Taylor model yielded good first-order agreement with the experiment.

Uniform strain assumption satisfies compatibility

conditions but not the stress equilibrium across grain

boundaries. Self-consistent (SC) approaches were developed to address the grain interaction in an average

sense. Molinari et al.

[7] presented a SC formulation for

texture predictions in large deformation of viscoplastic

polycrystals incorporating deformation heterogeneities.

Lebensohn and Tome ́

[8] extended the formulation to a

viscoplastic self-consistent (VPSC) model for the plastic

deformation in viscoplastic polycrystals, in which a

grain is treated as an inhomogeneity embedded in a

HEM that has the average property of the polycrystal.

Their model has been widely used for modeling material

anisotropic behaviors. Choi et al.[9] investigated the

macroscopic anisotropy in AA5019 sheets with both

Taylor and VPSC models, and the VPSC model showed

a better agreement with experimental data in yield

stresses and R values.

The advance of the computational power enables fullfield calculations of polycrystal constitutive behaviors.

Moulinec and Suquet[10] proposed a method based on

the fast Fourier transform (FFT) algorithm to compute

the local and overall responses of nonlinear composites.

Lebensohn[11] later adapted the FFT approach for

viscoplastic polycrystals. Sampling data in a regular

Fourier grid, the VPFFT model allows the response of

every individual site to vary independently. Delannay

et al.

[12] made quantitative comparisons of texture

L. HU, Graduate Student, and A.D. ROLLETT, Professor, are with

the Department of Materials Science and Engineering, Carnegie Mellon

University, Pittsburgh, PA 15213. Contact e-mail: linhu@andrew.

cmu.edu M. IADICOLA, Mechanical Engineer, T. FOECKE, Group

Leader, and S. BANOVIC, Materials Research Engineer, are with the

NIST Center for Metal Forming, National Institute of Standards and

Technology, Gaithersburg, MD 20899-8553.

Manuscript submitted July 15, 2010.

Article published online November 2, 2011

854—VOLUME 43A, MARCH 2012 METALLURGICAL AND MATERIALS TRANSACTIONS A
predictions from several plasticity models and concluded that N-site models give improved predictions

over Taylor-type models.

Whereas texture evolution is predominantly determined by the crystallographic nature of the available

slip systems, no ‘‘first principles’’ model exists for strain

hardening. Therefore, constitutive equations must be

developed by fitting to experimental data. Throughout

the years, constitutive relations based on crystal plasticity were well established for uniaxial loading, and

comparisons between forming limit data and multiaxial

numerical predictions were made. However, there is

little discussion in the literature on predicting both

stress-strain curves and texture development for processes that involve different loading paths for in-plane

and multiaxial deformation. Moreover, texture development often only has been analyzed in a qualitative

fashion.

In this study, the plastic response in AA5754 was

simulated with a full-constraint (FC) Taylor model,[4]

the tangent VPSC model,[8] and the N-site VPFFT

model,[11] and compared to multiaxial stress-strain

curves and texture evolutions from the experiment.

The objective of the study is to establish constitutive

equations that properly account for strain hardening

and grain (i.e., lattice) rotation during deformation. By

incorporating such crystal plasticity based equations

into plasticity models, the accuracy of predictions for

stress-strain curves and texture evolutions can be

improved.

II. EXPERIMENT

The material used in this investigation was industrially processed AA5754-O sheet, nominally 1-mm thick.

Its chemical composition is reported in Table I. The

alloy derives most of its strength from the solid solution

strengthening with magnesium atoms, interacting with

dislocations, and inhibiting dynamic recovery processes

during straining.[13]

The five strain paths investigated were equibiaxial

tension, uniaxial tension in the rolling direction (RD),

uniaxial tension transverse to the loading direction

(TD), plane strain with the first principal strain in the

RD (with near zero strain in the TD), and plane strain

with the first principal strain in the TD (with near zero

strain in the RD). The samples were tested in the plane

of the sheet. Uniaxial testing was performed using dogbone sheet tension specimens. Biaxial tension tests were

performed using a technique based upon the Raghavan

modification of the Marciniak biaxial stretching test.[14]

This technique holds the sheet sample in a binder while

the center area is stretched using a cylindrical ram with

an edge radius (to form a cup shape in the sample).

Nearly linear strain ratios from equibiaxial to uniaxial

(up to forming limit strains) can be achieved by reducing

the specimen width and thus the amount of constraint in

the reduced width direction. The presence of unquantifiable friction forces means that no simple relation

between ram force and applied stress exists. Therefore,

stresses are measured near the surface of the sheet

directly above the centerline of the ram by a modified

X-ray diffraction (XRD) residual stress measuring

system.[14] The strains were measured simultaneously

using a calibrated biaxial extensometer in contact with

the surface of the sheet located slightly off-center so as

to not interfere with the XRD stress measurement.

Uniaxial samples were tested with, and without, XRD

stress measurement, while using a calibrated uniaxial

extensometer and calibrated load cell to verify the XRD

measurements. The stress-strain data used in this article

was measured using the XRD technique exclusively in

order to produce a SC set.[15] Multiple tests were

performed for the plane strain and biaxial tests to verify

repeatability.

The textures of the samples were measured using

XRD technique.[16] Three incomplete pole figures were

measured with the tilt angle a ranging from 0 to 65 deg,

namely, {111}, {200}, and {220}. The full pole figures

and three-dimensional (3-D) orientation distribution

(OD, which gives the normalized probability density of

finding a given texture component specified by the three

Euler angles, u1, F, and u2) were derived from the

measured pole figures using the preferred orientation

package from Los Alamos (popLA).[17]

III. MODELING

Three polycrystal plasticity models were used to

simulate the AA5754 response under the five different

strain paths, namely, a FC Taylor model, a tangent

VPSC model, and the N-site VPFFT model. The

purpose of using the three different models was to

explore the sensitivity of the constitutive fitting and

texture development to the choice of model.

A. Single-Crystal Constitutive Behavior

All three models assume the same rate-sensitive

single-crystal constitutive behavior, Eq. [1].

e

0 1⁄4 c0

X

N

a1⁄41

ma ma : r

^sa

















K

sgn ma 1⁄21⁄2 : r 1

Here, e

0 and r are the plastic strain rate and stress, c0

is the reference shear rate, N is the number of slip

Table I. Chemical Composition of AA5754 Used in the Study

Element Mg Mn Fe Si Cr, Cu, Pb, Ni, Sn, Ti, Zn Aluminum

Weight percent 3.75 0.29 0.24 0.06 individually <0.05 balance

METALLURGICAL AND MATERIALS TRANSACTIONS A VOLUME 43A, MARCH 2012—855
systems, ^sa is the critical resolved shear stress (CRSS) of

the slip system, and j is the power-law exponent (inverse

of rate sensitivity). ma is the Schmid tensor for the slip

system a defined as

ma 1⁄4 1

2 ba  na þ na  ba ð Þ1⁄22

where b and n are the slip direction and slip plane

normal.

Equation [1] is an implicit equation that is solved to

find the stress state in each grain (or, at each Fourier

point in the VPFFT model) that generates the prescribed strain rate. This is done in all the models. The

skew-symmetric part of the slips is then used to update

the grain orientation, which allows the Taylor factor to

evolve with strain. As described in the next section, the

magnitude of the slip on each system is used to evolve

the CRSS during deformation in a grain. The physical

basis for the stress increase is mainly attributed to the

increase in dislocation density that occurs as dislocations on intersecting systems entangle with each other.

Since dislocations interact with various barriers to their

motion, such as other dislocations, grain boundaries,

and dislocation cell walls, the constitutive relation that is

fit to the data implicitly accounts for all contributions to

flow stress. In this work, we only consider the case of

isotropic hardening; i.e., each slip system hardens at the

same rate. If strain path changes are considered, then

more complex models may be required such as latent

hardening.

B. Strain Hardening

The Voce type strain hardening is adopted in the three

models in slightly different forms. For the FC Taylor

model, a strain-hardening rate h describes the evolution

of CRSS s from an initial CRSS, s0, toward a saturation

value sv:

h 1⁄4 ds

dC 1⁄4 h0 1  ssv

  1⁄23

where h0 is the initial hardening rate and C is the shear

on a given slip system. For the VPSC and n-site

VPFFT models, an extended Voce hardening rule governs the isotropic strain hardening behavior:

s 1⁄4 s0 þ s1 1  exp C h0

s1

    1⁄24

where (s0+s1) is the back-extrapolated CRSS.

(a)

(b)

(c) (d)

Fig. 1—Comparison of 0, 45, and 60 deg u2 sections in (a) the experimental OD and fitted ODs for the (b) LApp, VPSC, and (c) VPFFT models. Locations of ideal texture components (Cube, Brass, and Copper) are noted. The unit for contour levels is multiples of random intensity. (d)

The 128 3 128 3 128 synthetic microstructure for the VPFFT model with 5403 grains is shown. Colors denote different grains.

856—VOLUME 43A, MARCH 2012 METALLURGICAL AND MATERIALS TRANSACTIONS A
C. Grain Interaction

Assumptions on how grains interact link the effective

response of polycrystalline aggregates to individual

grain response. The three models have varying degrees

of complexity for interactions between grains, from

none in the Taylor-type models to all sites varying

independently in the VPFFT model.

The FC Taylor model assumes a uniform strain on all

grains. The applied macroscopic boundary condition is

imposed on all grains, and averaging the grain behaviors

gives the material response. The Taylor or Bishop–Hill

solution in the rate-insensitive limit provides an upper

bound approximate answer, which is then used as the

initial estimate of the (deviatoric) stress in Eq. [1].

The tangent VPSC model allows each grain to interact

with its surroundings in an average sense. The grain is

treated as an ellipsoidal inclusion embedded in a

homogeneous effective medium (HEM), the properties

of which are not known a priori and are adjusted selfconsistently to coincide with the average of all grains.

Under the HEM assumption, the strain and stress in

each grain varies as a function of orientation and the

stiffness of the interaction, depending on the choice of

homogenization methods for approximating a linear

Table II. Comparison of Cube, Brass, and Copper Volume

Fractions; Euler Angles (Degrees) for Each Component

Are Listed

Cube

(0,0,0)

Brass

(35,45,0)

Copper

(90,35,45)

Uniform texture 0.0067 0.012 0.013

Experiment 0.024 0.027 0.022

LApp/VPSC 0.021 0.027 0.022

VPFFT 0.021 0.022 0.018

Fig. 2—Error vs the initial CRSS s0. A single minimum of the total

error is found at s0 = 29 MPa.

Fig. 3—3-D plot of the error vs the initial hardening rate h0 and

CRSS s1. The error values are projected on the bottom and plotted

with contour levels. A single minimum is found at {h0, s1} = {310,

96}, noted by the star.

Table III. Strain Hardening Parameters Fitted

for the Plasticity Models

s0 (MPa) s1 (MPa) sv (MPa) h0 (MPa)

LApp 29 — 82 210

VPSC 33 96 — 310

VPFFT 32 80 — 320 Fig. 5—Comparison of measured and simulated stress-strain curves

for TD uniaxial tension.

Fig. 4—Comparison of measured and simulated stress-strain curves

for RD uniaxial tension.

METALLURGICAL AND MATERIALS TRANSACTIONS A VOLUME 43A, MARCH 2012—857
behavior at grain level. The tangent approximation used

in this study is obtained by a first-order Taylor

expansion of the constitutive equation in the vicinity

of overall response. It was shown that, opposed to the

completely stiff body assumed in the Taylor model, the

tangent scheme described a relatively compliant matrix

in terms of the grain to matrix interaction.[18]

Whereas the VPSC model assumes homogeneous

deformation inside the grains, the N-site VPFFT model

takes into account the interaction of each point with all

the other points in the (periodic) simulation volume. It

provides a full-field solution to both the equilibrium and

compatibility equations.

IV. SIMULATIONS

To simulate the stress-strain response and texture

evolutions in AA5754, three plasticity codes were used,

including the Los Alamos polycrystal plasticity (LApp)

code based on the FC Taylor model,[4] the VPSC code

developed by Lebensohn and Tome ́,

[8]

and the 3-D N-site

VPFFT code developed by Lebensohn.[11] A weighted

list of orientations was fitted to the measured initial

texture as the input for the simulations. The material

parameters used in the LApp and VPSC simulations

were identified by the best fit (calculated with a cost

function that sums the root-mean-square (RMS) differences between points) to the experimental values based

on the stress-strain curves in the five mechanical tests

being equally weighted in the error calculation. A simple

0 

50 

100 

150 

200 

250 

300 

0 5 10 15 

True Stress [MPa] 

True Strain [%] 

Plane Strain (TD) 

Exp (RD) Exp (TD) LApp VPSC VPFFT 

TD

RD

Fig. 7—Comparison of measured and simulated stress-strain curves

for TD-oriented plane strain tension. The horizontal axis is the true

strain in TD, and stresses both in RD and TD are plotted in the

graph.

0 

50 

100 

150 

200 

250 

300 

350 

0 5 10 15 

True Stress [MPa] 

True Strain [%] 

Plane Strain (RD) 

Exp (RD) Exp (TD) LApp VPSC VPFFT 

RD

TD

Fig. 6—Comparison of measured and simulated stress-strain curves

for RD-oriented plane strain tension. The horizontal axis is the true

strain in RD, and stresses both in RD and TD are plotted in the

graph.

0 

(a)

(b)

50 

100 

150 

200 

250 

300 

350 

0 

50 

100 

150 

200 

250 

300 

350 

0 5 10 15 20 

0 5 10 15 20 

True Stress [MPa] True Stress [MPa] 

True Strain [%] 

True Strain [%] 

Equibiaxial Stretch (RD) 

Experiment LApp 

VPSC VPFFT 

Fig. 8—Comparison of measured and simulated (a) RD and (b) TD

stress-strain curves for equibiaxial stretch.

Fig. 9—Bar chart for the quality of fit to the experimental data in

the five strain paths for the three different viscoplasticity models.

The error value in the vertical axis is the RMS difference between

the simulated stress-strain curves and the experimental data.

858—VOLUME 43A, MARCH 2012 METALLURGICAL AND MATERIALS TRANSACTIONS A
test grid of parameter values was used because of the

small number of parameters to be fitted, which has the

advantage of confirming that a global minimum exists in

the cost function. The VPFFT simulation is significantly

more time-consuming than LApp or VPSC, so the set of

hardening parameters was manually selected, starting

with the values determined for VPSC.

A. Fitting the Initial Texture

The fitting of the measured initial texture is required

since the polycrystal is represented by weighted orientations in the plasticity modeling. Figure 1(a) presents

the OD sections derived from the experimental pole

figures of as-received AA5754-O sheet with ideal texture

components noted on the graphs. The sheet metal

samples are weakly textured with a combination of

recrystallization (Cube) and retained deformation (Brass

and Copper) components.

The experimental OD was used to generate a weighted

set of 24,938 discrete orientations that are representative

of the texture. The discrete texture was used as the

starting texture in the LApp and VPSC simulations. The

weighted orientations after the plasticity simulation

were rebinned into the OD space applying both cubic

crystal symmetry and orthorhombic sample symmetry.

The fitted texture is shown in Figure 1(b). In the VPFFT

model, the polycrystalline microstructure takes the form

of a representative volume element (RVE) with periodic

boundary conditions and the RVE is discretized into a

Fig. 10—Comparison of OD sections for RD uniaxial tension. Strain levels are 5, 10, and 15 pct in RD.

METALLURGICAL AND MATERIALS TRANSACTIONS A VOLUME 43A, MARCH 2012—859
regular grid of points. A synthesized 128 9 128 9 128

microstructure of 5403 equiaxed grains, as shown in

Figure 1(d), was built using the Microstructure

Builder.[19] Fewer grains (than for the LApp and VPSC

models) were used to avoid excessively long computing

times. The equiaxed grain structure corresponds to the

grain shapes observed experimentally.[16]

Orientations of

grains were assigned using an optimization technique to

resemble the initial texture. The fitted texture for the

VPFFT model is presented in Figure 1(c).

A good match between the measured and fitted

textures can be seen from the comparison. To quantify

the fitting, volume fractions of major components were

calculated assuming a 10 deg orientation spread, as

listed in Table II. A 5 deg discretization of the OD space

was used. Volume fractions for a random (uniform)

texture are also included as a reference. The fitted

texture well captured major texture components, especially for Cube, the intensity of which was almost

4 times random. The fitting was better for the LApp and

VPSC models than the VPFFT model because of the

smaller number of grains used in the latter model.

B. Strain Hardening Parameter Fitting

The strain hardening parameters to be identified are

the initial CRSS, s0, the initial hardening rate, h0, and

the saturation CRSS, sv, in the FC Taylor model (or s1

in the VPSC and VPFFT models), as in the hardening

Eqs. [3] and [4]. Since the experimental data include

Fig. 11—Comparison of OD sections for TD uniaxial tension. The strain levels are 5, 10, and 15 pct in TD.

860—VOLUME 43A, MARCH 2012 METALLURGICAL AND MATERIALS TRANSACTIONS A
five different strain paths (the RD uniaxial tension, the

TD uniaxial tension, the RD plane strain, the TD

plane strain, and the equibiaxial stretch), the objective

was to determine a single set of hardening parameters

to match the five stress-strain curves. The underlying

assumption is that it is valid to use a single constitutive

law at the single slip system level. To find the best

match of the model predictions with the experimental

data, we searched a range of hardening parameter

values, sampling the parameter space on a regular grid

to ensure that a unique minimum of the error value

was identified. Although more expensive than, for

example, a steepest descent method, the small number

of parameters means that the complete set of simulations only requires a few hours on a standard desktop

and ensures that the minimum in the sampled parameter value range exists.

The error of a simulated stress-strain curve is calculated by the RMS difference from the experimental data:

error 1⁄4

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Pn

i1⁄40

Ssimul

i  Sexp

i

 2

n þ 1

vuuut 1⁄25

where n is the strain steps at which the measurement was

taken, Ssimul is the stress value from the model, and Sexp

is the measured stress value at the corresponding strain

level. The total error for a given set of parameter values

is the average of the error of the five stress-strain curves.

Fig. 12—Comparison of OD sections for RD-oriented plane strain tension. The strain levels are 5, 10, and 15, in RD.

METALLURGICAL AND MATERIALS TRANSACTIONS A VOLUME 43A, MARCH 2012—861
The stress at the zero plastic strain level is influenced

only by the texture and the initial CRSS s0. In the

simulation, it is independent of the initial hardening

rate and the saturation CRSS since no strain hardening

has yet occurred. Therefore, we can use the stresses at

zero plastic strain to obtain the value of s0. The error

value is obtained by setting n = 0 in Eq. [5], and thus,

only the data points at zero plastic strain are included.

Figure 2 shows the evolution of error values as s0

changes from 20 to 38 MPa with an increment of

1 MPa for the LApp simulation as an example. Each of

the five stress-strain curves is associated with a minimum in error at a value of initial CRSS that does not

necessarily coincide with the value corresponding to

minimum in total error. However, averaging the five

individual errors does give a single minimum in the

tested range (with no evidence of local minima), as

shown by the solid line in Figure 2, and an optimal

s0 = 29 MPa was identified, close to the individual

minima for uniaxial and equibiaxial minima. The

minimum error point lies at smaller strains, however,

for the plane strain loading paths and corresponds to a

shallower minimum in both cases.

The other two strain hardening parameters (h0 and s1

for the VPSC simulation in the example) are obtained in

the same fashion. Error values are calculated by varying

h0 and s1 simultaneously, and a best set of hardening

parameters is found at the minimum, as shown in

Figure 3. The optimal set of hardening parameters is

found at {h0, s1} = {310, 96}.

Fig. 13—Comparison of OD sections for TD-oriented plane strain tension. The strain levels are 5, 10, and 15 pct in TD.

862—VOLUME 43A, MARCH 2012 METALLURGICAL AND MATERIALS TRANSACTIONS A
Parameters used in the three models are listed in

Table III. Error values were also calculated in terms of

absolute difference and CHI-square difference between

the simulation and the experiment, but the resulting

hardening parameter values were negligibly different

from those fitted with the RMS error. It is also evident

that the different models yielded similar parameter

values as expected from the use of a similar law for all

three codes.

V. RESULTS AND DISCUSSION

A. Stress-Strain Curve Predictions

Figures 4 through 8 depict the stress-strain curves for

AA5754 in the aforementioned five deformation modes

simulated by LApp, VPSC, and VPFFT as well as those

determined experimentally. The error bars shown for the

experimental data are based on the measurement uncertainties determined for each data point individually, as

Fig. 14—Comparison of OD sections for equibiaxial stretch. The strain levels are 5, 10, 15, and 20 pct in RD (or TD, since strains in the two

directions are the same).

Table IV. Bunge Euler Angles for Texture Components Along the b-Fiber (from Copper to Brass)

u1 (deg) 90 80 70 65 60 55 50 45 40 35

F (deg) 35 35.56 36.11 36.67 37.22 37.78 38.33 42.78 43.89 45

u2 (deg) 45 50 55 60 65 70 75 80 85 90

METALLURGICAL AND MATERIALS TRANSACTIONS A VOLUME 43A, MARCH 2012—863
described in Iadicola et al.

[15] The largest average

experimental uncertainties seen are associated with the

equibiaxial data (±10 MPa), and the smallest values are

associated with the uniaxial data (±6 MPa). Looking at

the simulation predictions, overall better fits are seen in

VPSC and VPFFT than for the FC Taylor model. To

summarize the differences quantitatively, the RMS

errors associated with the quality of the fit to the

experimental data are given in Figure 9 for each strain

path and each model. The error value on the vertical

axis is the RMS difference between the simulated stressstrain curves and the experimental data for each case.

VPSC and VPFFT models give smaller error values

(therefore better qualities of predictions), compared to

the FC Taylor model.

With the same isotropic strain hardening behavior

adopted in the three models, the better fitting from the

VPSC and VPFFT models suggests that incorporating

the grain interaction improves the modeling of plastic

deformation. However, for the two plane strain tension

cases, for which stress was measured in two perpendicular directions, the VPSC and VPFFT models are often

only able to closely match one of the two stresses,

resulting in a root mean error larger than the average

experimental uncertainty. The nontrivial deviation of

calculated stresses from experimental data suggests the

need to further analyze the strain hardening behavior or

perhaps a more sophisticated hardening model. As far

as the authors are aware, this represents the first time

that plane strain experimental data of this type (that do

not require specimen bending or the assumptions

associated with a finite element model to determine

the stress values) were used to develop constitutive

relations.

0 

1 

2 

3 

4 

Volume Fraction [M.R.D.] 

2

0 

1 

2 

3 

4 

45 55 65 75 85 45 55 65 75 85 

Volume Fraction [M.R.D.] 

2

0 

1 

2 

3 

4 

Volume Fraction [M.R.D.] 

2

0 

1 

2 

3 

4 

45 55 65 75 85 45 55 65 75 85 

Volume Fraction [M.R.D.] 

2

0 strain 

5% strain 

10% strain 

15% strain 

Copper Brass Copper Brass 

Copper Brass Copper Brass 

Experiment LApp 

VPFFT

(d)

VPSC

(c)

(a) (b)

Fig. 15—b-fiber volume fraction plots for the RD uniaxial tension from (a) the experiment, (b) the LApp simulation, (c) the VPSC simulation,

and (d) the VPFFT simulation.

864—VOLUME 43A, MARCH 2012 METALLURGICAL AND MATERIALS TRANSACTIONS A
B. Texture Evolution

Figures 10 through 14 demonstrate the development

of OD as a function of deformation mode and strain

level. 0, 45, and 60 deg u2 sections in OD from

experiments and simulations are compared at the same

strain levels stated as follows: 5, 10, and 15 pct for

uniaxial tension and plane strain, and 5, 10, 15, and

20 pct for equibiaxial stretch.

The following observations are made from the experimental ODs. For RD uniaxial tension, the strain is not

large enough for the material to form complete h100ifiber and h111i-fiber. The material starts with a b-fiber

texture, and components along the b-fiber parallel to

h100i or h111i directions intensify. For TD uniaxial

tension, grains tend to rotate to the Cube and Brass

orientations. The b-fiber further develops in the RD

plane strain. Grains move toward the Brass orientation

in TD-oriented plane strain. h110i crystal axes gradually

aligned with the sample ND in the equibiaxial stretch,

forming a h110i-fiber texture. Qualitatively, the three

plasticity models predict the same patterns, except for

the equibiaxial stretch, where the FC Taylor model fails

to predict the removal of cube orientations.

The simulated evolution of texture intensities varies

markedly among the models. To quantify the texture

evolution, volume fractions along the beta fiber are

calculated for uniaxial tension and plane strain cases.

Table IV lists the Bunge Euler angles for the ten points

chosen along the fiber. Since the development of a (110)-

fiber is most prominent in the equibiaxial stretch test,

orientations along the (110)-fiber (u1 = 0 to 35 deg,

F = 45 deg, and u2 = 0 deg) are selected instead for

this strain state. The fiber volume fraction plots are

shown in Figures 15 through 19. We can see that, in

0 

1 

2 

3 

4 

Volume Fraction [M.R.D.] 

2

0 

1 

2 

3 

4 

5 

45 55 65 75 85 45 55 65 75 85 

Volume Fraction [M.R.D.] 

2

0 

1 

2 

3 

4 

Volume Fraction [M.R.D.] 

2

0 

1 

2 

3 

4 

45 55 65 75 85 45 55 65 75 85 

Volume Fraction [M.R.D.] 

2

0 strain 

5% strain 

10% strain 

15% strain 

Copper Brass Copper Brass 

Copper Brass Copper Brass 

Experiment LApp 

VPSC VPFFT

(c) (d)

(a) (b)

Fig. 16—b-fiber volume fraction plots for the TD uniaxial tension from (a) the experiment, (b) the LApp simulation, (c) the VPSC simulation,

and (d) the VPFFT simulation.

METALLURGICAL AND MATERIALS TRANSACTIONS A VOLUME 43A, MARCH 2012—865
general, the VPSC simulation results in volume fractions

much higher than experiment (note the change in scale

for the VPSC results), while the LApp simulation is

quite reliable for uniaxial and plane strain cases, and the

VPFFT model appears to capture more of the overall

development in most cases.

The VPSC model we used in the study is essentially a

one-site model. The grain interaction is dealt with in an

average sense and no special treatment is given to the

local environment. Therefore, grains with the same

orientation, but different neighbors, will deform in a

same way. It is similar to the FC Taylor model for

texture simulation in the sense that grains with the same

initial orientations will rotate in the same fashion,

leading to orientation concentrations and thus higher

predicted texture intensities than those seen in the

experiment.[20] A major difference between the FC

Taylor model and the tangent one-site VPSC model is

the assumption on the grain interaction. The FC Taylor

model assumes a completely stiff body, while the tangent

VPSC model assumes a relatively compliant grain to

matrix interaction. Therefore, it appears that the faster

texture evolution seen in the VPSC simulation is due to

its overestimate of the matrix compliance. On the other

hand, it is worth mentioning that the FC Taylor model

fails to predict the removal of cube orientations in

equibiaxial tension, while the VPSC model captures the

correct texture evolution. This finding indicates that the

actual interaction behavior might be between the complete stiffness assumed in the FC Taylor model and the

tangent behavior. Intermediate interaction schemes

include stiffer inclusion to matrix formulations such as

0 

1 

2 

3 

4 

Volume Fraction [M.R.D.] 

2

0 

1 

2 

3 

4 

45 55 65 75 85 45 55 65 75 85 

Volume Fraction [M.R.D.] 

2

0 

1 

2 

3 

4 

Volume Fraction [M.R.D.] 

2

0 

1 

2 

3 

4 

45 55 65 75 45 55 65 75 85 

(a) (b)

(c) (d)

85 

Volume Fraction [M.R.D.] 

2

0 strain 

5% strain 

10% strain 

15% strain 

Copper Brass Copper Brass 

Copper Brass Copper Brass 

Experiment LApp 

VPSC VPFFT

Fig. 17—b-fiber volume fraction plots for the RD-oriented plane strain from (a) the experiment, (b) the LApp simulation, (c) the VPSC simulation, and (d) the VPFFT simulation.

866—VOLUME 43A, MARCH 2012 METALLURGICAL AND MATERIALS TRANSACTIONS A
the secant[21]

and affine[22]approaches. The prediction of

a more pronounced texture by the tangent VPSC model

is also reported by M’Guil et al.

[23] for deformation

texture in tension and compression and also by Li

et al.[20]

for the texture evolution in copper during equal

channel angular extrusion. It has been further shown

that to weaken the texture evolution prediction in the

VPSC model, grain-grain interaction needs to be

accounted for in more detail, such as incorporating a

grain co-rotation scheme[24] for orientation correlations

or implementing a rule for grain subdivisions.[25]

The improvement by the N-site VPFFT model is

largely due to the fact that voxels in a grain are allowed

to deform differently such that internal orientation

gradients can develop within grains. This effectively

increases the spread in grain orientations and retards the

texture development.

C. Lankford Coefficient Comparison

As a further check of the quality of the simulations, a

comparison is made between predicted and measured

values of the Lankford coefficient. Otherwise known as

the R value, it measures the plastic anisotropy in the

material. It is defined as the ratio of the width plastic

strain to the through-thickness plastic strain in a uniaxial

tensile test. Using the material parameters we obtained

from the stress-strain curve fitting, we were able to

compute R values for the tested material, as shown in

Figure 20. The R values were calculated with both the

LApp code and the tangent VPSC code, at an angular

interval of 15 deg (the use of stress boundary conditions

on the overall polycrystal in the VPFFT code was not

implemented, which is what is required for the R-value

calculation). When compared to the experimental values

0 

1 

2 

3 

4 

Volume Fraction [M.R.D.] 

2

0 

1 

2 

3 

4 

5 

6 

Volume Fraction [M.R.D.] 

2

0 

1 

2 

3 

4 

Volume Fraction [M.R.D.] 

2

0 

1 

2 

3 

4 

45 55 65 75 85 45 55 65 75 85 

45 55 65 75 85 45 55 65 75 85 

Volume Fraction [M.R.D.] 

2

0 strain 

5% strain 

10% strain 

15% strain 

Copper Brass Copper Brass 

Copper Brass Copper Brass 

Experiment LApp 

VPSC VPFFT

(c) (d)

(a) (b)

Fig. 18—b-fiber volume fraction plots for the TD-oriented plane strain from (a) the experiment, (b) the LApp simulation, (c) the VPSC simulation, and (d) the VPFFT simulation. Note the scale change in (c) the VPSC plot.

METALLURGICAL AND MATERIALS TRANSACTIONS A VOLUME 43A, MARCH 2012—867
in Figure 20, the calculated values show relatively good

agreement, particularly considering that no parameter

adjustments were made in the model to optimize the fits

to compare to the Lankford coefficient. Bate and An[26]

compared the measured R value in AA5005 with the

Taylor model predictions and showed similar deviations.

VI. CONCLUSIONS

Plastic deformation under various strain paths for

AA5754 aluminum alloy was simulated with the FC

Taylor model, a tangent VPSC model, and the N-site

VPFFT model. The single slip system hardening parameters were fitted by comparison with the experimental

stress-strain curves. It was shown that allowing stress

and strain variations in grains improved multiaxial

0 

1 

2 

3 

4 

Volume Fraction [M.R.D.] 

1

0 

2 

4 

6 

8 

10 

0 10 20 30 0 10 20 30 

Volume Fraction [M.R.D.] 

1

0 

1 

2 

3 

4 

Volume Fraction [M.R.D.] 

1

0 

1 

2 

3 

4 

0 10 20 30 0 10 20 30 

Volume Fraction [M.R.D.] 

1

0 strain 

5% strain 

10% strain 

15% strain 

20% strain 

Goss Brass Goss Brass

Goss Brass Goss Brass

Experiment LApp 

VPFFT

(d)

VPSC

(c)

(a) (b)

Fig. 19—<110>-fiber (u1 = 0 to 35 deg, F = 45 deg, and u2 = 0 deg) volume fraction plots for equibiaxial stretch from (a) the experiment, (b)

the LApp simulation, (c) the VPSC simulation, and (d) the VPFFT simulation. Note the scale change in (c) the VPSC plot.

Fig. 20—Calculated R values vs the angle between the rolling direction and the stressing direction, with experimental values and error

bars for comparison.

868—VOLUME 43A, MARCH 2012 METALLURGICAL AND MATERIALS TRANSACTIONS A
stress-strain predictions. Texture development was also

compared with experiments both qualitatively and

quantitatively. Overall, the N-site VPFFT model had

the best texture prediction. The results indicate that

taking grain interaction into consideration refines the

polycrystal plasticity modeling. However, the failure to

predict the mechanical response equally well for all the

different multiaxial loading paths points out the need to

find more effective ways to describe accurately the strain

hardening.

ACKNOWLEDGMENTS

This work was supported at Carnegie Mellon University by the National Institute of Standards & Technology under Award No. 70NANB8H8038. Use of

facilities provided by the Higher Education Program,

Intel Corporation, and by the MRSEC, CMU, under

NSF Grant No. DMR-0520425 is also gratefully

acknowledged. Use of the VPFFT code from Ricardo

Lebensohn, Los Alamos National Laboratory, is

gratefully acknowledged.

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