An Extended Age-Hardening Model for Al-Mg-Si Alloys Incorporating the Room-Temperature Storage and Cold Deformation Process Stages

Abstract

In this article, a new age-hardening model for Al-Mg-Si alloys is presented (named NaMo-Version 2), which takes into account the combined effect of cold deformation and prolonged room-temperature storage on the subsequent response to artificial aging. As a starting point, the original physical framework of NaMo-Version 1 is revived and used as a basis for the extension. This is permissible, since a more in-depth analysis of the underlying particle-dislocation interactions confirms previous expectations that the simplifying assumption of spherical precipitates is not crucial for the final outcome of the calculations, provided that the yield strength model is calibrated against experimental data. At the same time, the implementation of the Kampmann–Wagner formalism means that the different microstructure models can be linked together in a manner that enforces solute partitioning and competition between the different hardening phases which form during aging (e.g., clusters, β″ and β′). In a calibrated form, NaMo-Version 2 exhibits a high degree of predictive power, as documented by comparison with experiments, using both dedicated nanostructure and yield strength data as a basis for the validation. Hence, the model is deemed to be well-suited for simulation of thermomechanical processing of Al-Mg-Si alloys involving cold-working operations like sheet forming and stretch bending in combination with heat treatment and welding.

Full text

A Combined Precipitation, Yield Strength, and Work
Hardening Model for Al-Mg-Si Alloys
OLE RUNAR MYHR, ØYSTEIN GRONG, and KETILL OLAV PEDERSEN
In the present article, a new two-internal-variable model for the work hardening behavior of
commercial Al-Mg-Si alloys at room temperature is presented, which is linked to the previously
developed precipitation and yield strength models for the same class of alloys. As a starting
point, the total dislocation density is taken equal to the sum of the statistically stored and the
geometrically necessary dislocations, using the latter parameters as the independent internal
variables of the system. Classic dislocation theory is then used to capture the overall stress-strain
response. In a calibrated form, the work hardening model relies solely on outputs from the
precipitation model and thus exhibits a high degree of predictive power. In addition to the solute
content, which determines the rate of dynamic recovery, the two other microstructure param-
eters that control the work hardening behavior are the geometric slip distance and the corre-
sponding volume fraction of nonshearable Orowan particles in the base material. Both
parameters are extracted from the predicted particle size distribution. The applicability of the
combined model is illustrated by means of novel process diagrams, which show the interplay
between the different variables that contribute to work hardening in commercial Al-Mg-Si
alloys.
DOI: 10.1007/s11661-010-0258-7
ÓThe Minerals, Metals & Materials Society and ASM International 2010
I. INTRODUCTION
THE age hardening Al-Mg-Si alloys undergo com-
plex structural changes during heat treatment. The
extent of these structural changes depends on the
applied temperature, the time the material is exposed
to temperature, and how fast it is cooled to room
temperature.
[1,2]
In the as-quenched condition, the
alloying elements are present in solid solution or in the
form of small clusters distributed throughout the matrix.
During prolonged room-temperature storage, the mate-
rial regains some strength as the clusters grow, but the
strength level saturates within a couple of days because
the diffusion processes responsible for the growth of
clusters are very slow at room temperature.
[3–5]
How-
ever, if the temperature is raised to, say, 423 K (150 °C)
to 473 K (200°C), the kinetics become faster and a high
fraction of small hardening precipitates forms, causing a
continuous increase in the strength level until the peak
of the hardness curve is reached.
[6–8]
Further heat
treatment then leads to overageing, meaning that the
alloy starts to lose strength. This strength loss is related
to a change in precipitate size and number density along
with a loss of matrix coherency during heat treatment,
which makes it easier for the dislocations to bypass the
precipitates by Orowan looping.
[3–9]
It is obvious that a constitutive model capable of
capturing the stress-strain behavior of age hardening
aluminum alloys must take into account the microstruc-
tural changes that occur during heat treatment as well as
how these changes affect the accumulation and annihila-
tion of dislocations during deformation.
[9,10]
There seems
to be general agreement that both shearable and nonshe-
arable particles will strongly influence the deformation
behavior of precipitation-hardened materials.
[11–14]
On
the other hand, the mechanisms behind this strengthening
and how the dislocations nucleate, interact, and eventu-
ally become stored around the precipitates are contro-
versial issues that have been keenly debated in the
scientific literature for many years.
[11–20]
For example, it
is still unclear whether the work hardening behavior of
Al-Cu and Al-Mg-Si-Cu alloys is best understood in the
framework of additional storage of geometrically neces-
sary dislocations based on the Ashby formalism,
[12]
or in
terms of the development of long-range internal (back)
stresses in the aluminum matrix.
[18–20]
Fortunately, this long-lasting discussion about the
underlying strengthening mechanisms has not stopped
the development of useful work hardening models for
engineering materials. The well-established one-internal-
variable models of Estrin,
[21,22]
Cheng et al.,
[10]
and
Poole and Lloyd
[23]
can either predict the stress-strain
behavior of single-phase materials or account for the
effect of hardening precipitates on the flow stress at
room temperature in a semiempirical manner. They
OLE RUNAR MYHR, Adjunct Professor, is with Research and
Technology Development, Hydro Aluminium, N-2831 Raufoss,
Norway, and Structural Impact Laboratory (SIMLab), Norwegian
University of Science and Technology, N-7491 Trondheim, Norway.
Contact e-mail: ole.runar.myhr@hydro.com ØYSTEIN GRONG,
Professor, is with the Department of Materials Science and Engineer-
ing, Norwegian University of Science and Technology, N-7491
Trondheim, Norway, and Structural Impact Laboratory (SIMLab),
Norwegian University of Science and Technology. KETILL OLAV
PEDERSEN, Senior Research Scientist, is with Department of
Metallurgy, SINTEF Materials and Chemistry, N-7465 Trondheim,
Norway, and Structural Impact Laboratory (SIMLab), Norwegian
University of Science and Technology.
Manuscript submitted June 16, 2009.
METALLURGICAL AND MATERIALS TRANSACTIONS A

have all their origins in the classical work hardening
models of Kocks
[24]
and Mecking and Kocks,
[25]
but are
simplified and structured in such a way that the
contribution from various metallurgical parameters
such as the grain size, the solute content, and the
volume fraction of shearable or nonshearable particles
to the work hardening behavior can be isolated and
dealt with separately. The Estrin model
[21,22]
also has the
advantage that it can be generalized and recast into a
two-internal-variable model, which makes it more flex-
ible and applicable to a variety of materials and complex
heat treatment problems. Still, a major challenge with
this two-internal-variable approach is to determine the
numerical values of the different parameters entering the
model. This is because several optima can occur when
standard calibration techniques are used. Therefore, a
special parameter calibration technique (known as the
evolution strategy) has been developed by Estrin
[21]
to
determine the numerous adjustable constants that need
to be fixed before a prediction can be made.
In the past, separate precipitation and yield strength
models for commercial Al-Mg-Si alloys have been
developed by two of the present authors.
[26–28]
These
are sufficiently relevant and comprehensive to be used as
a tool to predict the response of the alloys to multistage
thermal processing and welding from knowledge of
chemical composition and thermal history. Recently, the
models have been successfully employed in combination
with work hardening theory to establish a new internal
variable constitutive model for the finite element (FE)
simulation of local hot forming of Al-Mg-Si alloys.
[29]
The basic idea behind the approach of Myhr et al.
[26–28]
is to link the particle size distribution to the macroscopic
yield stress at room temperature by considering the
intrinsic resistance to dislocation motion due to particles
and solute atoms. This locus represents, in turn, the
starting point of the plastic flow curve in stress-strain
space.
The next step in establishing an overall through-
process model for age hardening Al-Mg-Si alloys would
be to utilize the key precipitate and solute parameters
extracted from the precipitation model to calculate the
full work hardening response at room temperature
following multistage thermal heat treatment and welding.
For example, when used in combination with an appro-
priate mechanical model, the predicted work hardening
curves provide a systematic basis for optimizing the
resulting load-bearing capacity of crash components
made from such alloys.
[30]
However, in order to capture
the complex interactions between solute atoms, precip-
itates, and dislocations in an adequate manner, a two-
internal-variable model should be targeted. At the same
time, its underlying structure should be simple enough to
allow all adjustable numerical constants to be fitted by
calibration against experimental tensile and compression
test data. These two conflicting requirements suggest an
analytical formulation of the work hardening problem,
along the lines indicated by Ashby.
[12]
The basic idea behind the Ashby approach
[12]
is to
split the total dislocation density into statistically stored
and geometrically necessary dislocations, which, in
turn, allows a separate treatment of each category of
dislocations. Because the effectiveness of the nonshea-
rable particles to store dislocations is conveniently
described by the geometric slip distance,
[12]
the density
of geometrically necessary dislocations is a quantity that
can be evaluated from the predicted particle size
distribution. On the other hand, the density of the
statistically stored dislocations depends on the rate of
dynamic recovery, and can be calculated from the Kocks
evolution equation.
[24,25]
A requirement is, however,
that proper corrections are made for the contribution
from alloying elements in solid solution.
The splitting of the total dislocation density into
statistically stored and geometrically necessary disloca-
tions makes the Ashby approach
[12]
particularly useful
in modeling the work hardening behavior of commercial
Al-Mg-Si alloys at room temperature following artificial
aging. This is because the transition from shearable to
nonshearable particles during heat treatment then can
be captured in a continuous manner by calculating their
contribution to the flow stress separately, based on
outputs from the previously developed precipitation
and yield strength models.
[26–28]
Moreover, through
proper scaling and calibration of the constitutive equa-
tions entering the work hardening model, the errors
introduced by neglecting possible mutual interactions
between the two categories of dislocations can be
reduced. This scaling also makes the assumption of a
given particle shape less critical, as the geometrical slip
distance for both spherical particles and platelike
precipitates depends on the same microstructural
parameters (i.e., radius/plate thickness and volume
fraction),
[12]
which are calculated from the precipitation
model.
[26,27]
Finally, the use of a simple linear summa-
tion law for the different strength contributions in the
work hardening model (as adopted from Ashby
[12]
)
means that it can be completely separated from the yield
strength model and implemented without introducing a
new variable exponent in the expression for the flow
stress, as frequently done in the literature.
[10,31]
Due to the lack of a unified theory of dispersion
hardening in metals and alloys, it is obvious that, at
least, some of these simplifying assumptions and short-
cuts will be controversial and in conflict with other
people’s views and discretions. Therefore, in the context
of the present work hardening model, their justification
relies on a good correlation between predictions and
experiments.
II. COMPONENTS OF THE MODEL
The symbols and units used throughout the article are
defined in the Appendix. In the following, the different
components of the work hardening model will be
presented, starting with a brief review of the integrated
precipitation and yield strength models.
A. Precipitation Model
The precipitation model by Myhr and Grong
[26]
is the
key component in both the yield strength and the work
METALLURGICAL AND MATERIALS TRANSACTIONS A

hardening models, as illustrated in Figure 1. The model
consists of the following three components:
(1) a nucleation law, which predicts the number of sta-
ble nuclei that form at each time-step;
(2) a rate law, which calculates either the dissolution
or the growth rate of each discrete particle size
class; and
(3) a continuity equation, which keeps a record of the
amount of solute being tied up as precipitates.
Details of the underlying assumptions as well as a
description of the basic features of the model and the
solution algorithm used to capture the evolution of the
particle size distribution with time and temperature have
been reported elsewhere.
[26–28]
Hence, only a brief
summary of the main constitutive equations is given
subsequently.
The nucleation equation predicts the number of stable
nuclei that form at each time-step. If the incubation
period is neglected, the steady-state nucleation rate jis
conveniently expressed as:
[26,32]
j¼j0exp A0
RT

31
lnðC=CeÞ

2
"#
exp Qd
RT

½1
where the parameters j
0
,Q
d
, R, and Thave their usual
meaning, and are defined in the Appendix. Moreover, C
is the mean solute content in the matrix, and C
e
is the
equilibrium solute content at the particle/matrix inter-
face (given by the phase diagram). Note that the
parameter A
0
has the same dimension as the activation
energy for nucleation (i.e., J/mol) and is a measure of
the potency of the heterogeneous nucleation sites in the
parent material.
When a particle with radius rand solute concentra-
tion C
p
is introduced into the system, it will either
dissolve or grow, depending on whether the particle/
matrix interface concentration C
i
exceeds C. The rate at
which this occurs can be expressed as:
[33]
dr
dt ¼CCi
CpCi
D
r½2
where Dis the diffusion coefficient. The interface
concentration C
i
is, in turn, inter-related to the equilib-
rium concentration C
e
and the particle radius rthrough
the Gibbs–Thomson equation.
[26]
Finally, the mean solute content in the matrix Cis
calculated from the continuity equation to obtain the
instantaneous values of the nucleation and growth rates
in the precipitation model:
C¼C0CpC

X
i
4
3pr3
iNi½3
where C
0
is the initial solute content in the alloy and N
i
is the number of particles within the discrete radius
interval r
i
.
B. Yield Strength Model
The yield strength model converts the relevant output
parameters from the precipitation model into an equiv-
alent room-temperature yield stress through dislocation
mechanics by considering the following contribu-
tions:
[27,28]
(1) precipitation hardening due to shearing and
bypassing of particles by dislocations r
p
(in this
case, the Friedel formalism
[34]
is used to calculate
the mean planar particle spacing ‘along the bend-
ing dislocation in the expression for r
p
, based on
the methodology described in Reference 35); and
(2) solid solution hardening effects r
ss
, which include
the three strengthening elements Si, Mg, and Cu.
In alloys, in which several strengthening mechanisms
are operative at room temperature, it is reasonable to
assume that the individual strength contributions can be
added linearly.
[36]
Thus, taking r
i
equal to the intrinsic
yield strength of pure aluminum, the resulting expres-
sion for the overall macroscopic yield strength r
y
becomes
[27,28]
ry¼riþrss þrp½4
C. Work Hardening Model
In the work hardening model presented subsequently,
the total dislocation density q
t
is taken equal to the sum
of the geometrically necessary dislocations q
g
and
the statistically stored dislocations q
s
. According to
Ashby,
[12]
the geometrically necessary dislocations will
dominate if the geometric slip distance k
g
is smaller than
the corresponding slip distance for statistical storage of
dislocations k
s
, as measured from the length of the slip
lines in the work-hardened material. In the other extreme,
when k
g
>k
s
, their contribution to the total dislocation
density becomes negligible and can be ignored.
Fig. 1—Schematic diagram showing the coupling between the precip-
itation, yield strength, and work hardening models developed for
Al-Mg-Si alloys.
METALLURGICAL AND MATERIALS TRANSACTIONS A

In order to establish a useful work hardening model
for commercial Al-Mg-Si alloys, it is necessary to
convert the current values of the dislocation densities
into an equivalent room-temperature flow stress r.
Because the Ashby approach
[12]
is based on the assump-
tion that the two dislocation densities can be added
linearly in stress-strain space, their total contribution to
the net flow stress Dr
d
=r–r
y
is given by the following
equation:
Drd¼rry¼aMGbffiffiffiffiffi
qt
p¼aMGbffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
qsþqg
p½5
where ais a constant with a numerical value close to 0.3,
Mis the Taylor factor, Gis the shear modulus, and bis
the magnitude of the Burgers vector.
The form of Eq. [5] suggests that the work hardening
behavior of Al-Mg-Si alloys can be captured mathe-
matically by considering the net contribution from
dislocation hardening Dr
d
, using q
s
and q
g
as the
independent internal variables of the system.
1. Contribution from shearable particles and alloying
elements in solid solution
Previous experiments have shown that the work
hardening behavior of Al-Mg-Si alloys in the naturally
aged (T4) condition is essentially similar to that in the
as-quenched (W10) condition.
[37]
Hence, in the context
of the model developed, no distinction is made between
small shearable particles in the form of GP zones (or
clusters) and atoms in solid solution. This means that
the balance between statistical storage and dynamic
recovery of dislocations in both types of matrix mate-
rials is conveniently described by the evolution equation
proposed by Kocks:
[24,25]
dqs¼k1q1=2
sk2qs

dep½6
where k1is a constant being characteristic of the
material under consideration, whereas the k
2
parameter,
which determines the rate the dynamic recovery during
plastic deformation, depends on the actual solute
content in the alloy.
Note that this solute dependence is believed to arise
from multiple mechanisms, i.e., changes in the stacking
fault energy due to alloying elements and solute drag
effects on dislocations.
[24,25]
Hence, the k
2
parameter
would be expected to increase during artificial aging of
Al-Mg-Si alloys as the matrix becomes gradually
depleted with respect to Mg and Si.
In an integrated form, Eq. [6] reads:
qs¼k1
k2

2
1exp k2ep
2

2
½7
Thus, in the limiting case, in which the contribution
from geometrically necessary dislocations can be ig-
nored (i.e.,k
g
>k
s
), a combination of Eqs. [5] and [7]
leads to the following variant of the well-known Voce
equation:
Drd¼aMGbk1
k2
1exp k2ep
2

½8
The Voce equation is particularly useful in the
present context, because it yields a well-defined satu-
ration stress Drd¼Drd;sat large plastic strains e
p
,as
illustrated in Figure 2, although not all commercial
Al-Mg-Si alloys show this behavior.
[38]
At the same
time, it allows the underlying microstructural param-
eters k
1
and k
2
in the dislocation model to be evaluated
from an analysis of experimental tensile and compres-
sion test data. Thus, after the saturation stress Dr
d,s
is
reached, Eq. [8] yields the following expression for
the k
1
/k
2
ratio:
k1
k2¼Drd;s
aMGb½9
Moreover, by considering the form of the experimen-
tal work hardening curve, the individual values of k
1
and
k
2
for a given reference alloy can be determined on the
basis of simple curve fitting by selecting the best
combination that satisfies the constraint provided by
Eq. [9]. Finally, since k
1
is assumed to be constant for a
given family of alloys, the dependence of k
2
on the solute
content may be evaluated from tensile and compression
testing of a selected group of alloys yielding the required
span in the saturation stress Dr
d,s
in the as-quenched
(W10) condition.
2. Contribution from nonshearable Orowan particles
According to the definition of Ashby,
[12]
the geometric
slip distance k
g
is a measure of how far the dislocations
move before they are stored around particles that are
dispersed within the matrix material. This storage of
dislocations is necessary to obtain compatibility between
the two phases during deformation. The parameter
k
g
is therefore a characteristic of the microstructure
and is related to the type and distribution of the
hardening precipitates in the material. In general, the
density of geometrically necessary dislocations q
g
in a
Fig. 2—Work hardening behavior, as predicted from the Voce equa-
tion, yielding the characteristic saturation stress at large plastic
strains.
METALLURGICAL AND MATERIALS TRANSACTIONS A

dispersion-strengthened material is inter-related to the
geometric slip distance k
g
and the imposed shear strain c
through the following relationship:
[12]
qg¼4
b
c
kg½10
Equation [10] applies to materials containing different
types of nonshearable particles, e.g., equiaxed, spherical,
or platelike precipitates. Depending on their shape, size,
and volume fraction, the dislocation array required to
achieve compatible deformation can either be a stack of
prismatic loops, shear loops, or edge/screw dipoles.
[12]
However, because Eq. [10] is approximate and only
suitable for order of magnitude calculations, it cannot
readily be transferred to commercial engineering mate-
rials such as Al-Mg-Si alloys without further modifica-
tion. First, it needs to be calibrated against experimental
data in order to yield realistic values for q
g
during
plastic deformation. This is most easily done using a
simple normalizing (scaling) procedure by introducing a
chosen reference alloy, characterized by its geometric
slip distance of Orowan particles kg¼kg;o¼kref
g;o,and
a corresponding reference state defined by qg¼qref
g
when c=c
ref
, from which
qg¼qref
g
kref
g;o
kg;o
c
cref ½11
where k
g,o
specifically refers to the geometric slip
distance of hardening precipitates in Al-Mg-Si alloys
that fulfill the Orowan criterion.
The next step is to switch from shear strain cto tensile
strain eand then enforce an upper limit for the
dislocation density during deformation. The latter is
required because the accumulation of dislocation loops
around the precipitates (and, thus, q
g
), in practice,
cannot increase without limit, as predicted by Eq. [11].
In real engineering materials, the recovery mechanisms
preventing further accumulation of the geometrically
necessary dislocations will be triggered when the local
shear stress is high enough to either bring about
decohesion or fracture of the particles or to nucleate
new dislocations at the particle/matrix interface.
[12,20]
The net effect is that q
g
will saturate at some given
(critical) plastic strain e
p
=e
c
, as shown schematically
in Figure 3. This threshold strain depends, in turn, on
the volume fraction f
o
of the dispersed Orowan parti-
cles.
[12]
Taking e
c
=e
c
ref
when f
o
=f
o
ref
for the chosen
reference alloy, a rational expression for e
c
is:
ec¼fref
o
fo

eref
c½12
This leads to the following expressions for the density
of the geometrically necessary dislocations during plas-
tic deformation:
qg¼qref
g;s
kref
g;o
kg;o
ep
eref
c
when ep<ec½13
and
qg;s¼qref
g;s
kref
g;o
kg;o
ec
eref
c
when epec½14
where qref
g;sis the corresponding saturation value for q
g
in
the reference material at ec¼eref
c.
Finally, after substituting Eqs. [7], [13], and [14] into
Eq. [5], the appropriate relationships between Dr
d
,q
s
,
q
g
, and q
g,s
become:
Drd¼aMGbffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
k1
k2

2
1exp k2ep
2

2
þqref
g;s
kref
g;o
kg;o
ep
eref
c
s
when ep<ec½15
and
Drd¼aMGbffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
k1
k2

2
1exp k2ep
2

2
þqref
g;s
kref
g;o
kg;o
ec
eref
c
s
when epec½16
where the actual value of e
c
is given by Eq. [12].
3. Expressions for k
g,o
and f
o
Following calibration, the present work hardening
model relies solely on outputs from the precipitation
model. In addition to the mean matrix solute content C
(given by Eq. [3]), the two other microstructure param-
eters that control the work hardening behavior of
Al-Mg-Si alloys are the geometric slip distance k
g,o
and the corresponding volume fraction f
o
of nonshea-
rable Orowan particles in the matrix material. Both
parameters can be extracted from the predicted particle
size distribution, as shown schematically in Figure 4.
Let the Orowan particles be characterized by their
radius r
o
>r
c
, where r
c
is the critical particle radius for
dislocation bypassing, analogous to that done in the
yield strength model.
[27,28]
Then k
g,o
and f
o
are readily
calculated from the following expressions:
kg;o¼8X
r¼1
r¼ro
r2
iNi
!
1
½17
Fig. 3—Schematic illustration of the assumed qgeprelationship
during work hardening of Al-Mg-Si alloys. In this case, the density
of geometrically necessary dislocations is supposed to saturate when
the accumulation of dislocation loops around the precipitates ceases
at e
p
=e
c
.
METALLURGICAL AND MATERIALS TRANSACTIONS A

fo¼4p
3X
r¼1
r¼ro
r3
iNi½18
where N
i
is the number of particles per unit volume
within the size class r
i
.
Note that Eq. [17] correctly reduces to kg;o¼ro=foin
the limiting case when i= 1. This is in agreement with
Ashby’s definition of the geometric slip distance for a
homogeneous distribution of Orowan particles of radius
r
o
or plates with thickness D=r
o
.
[12]
III. CALIBRATION AND VALIDATION
OF WORK HARDENING MODEL
The calibration and subsequent validation of the
work hardening model are done using experimental
tensile and compression test data obtained from a series
of experiments carried out at the Norwegian University
of Science and Technology (NTNU), Department of
Materials Science and Engineering (DMSE, Trondheim,
Norway), and at the Hydro R & D Materials Technol-
ogy (Sunndalsøra, Norway) by Dr. Øyvind Ryen. Since
these results have been reported elsewhere,
[39,40]
only a
brief summary of the experimental program is given in
Section A.
A. Materials and Experimental Procedure
The chemical composition of the two DC-cast
AA6082 and AA6063 billet materials used in the
investigation is listed in Table I. In the as-homogenized
condition, both alloys reveal an equiaxed (recrystallized)
grain structure with a mean grain size of about 80 to 100
lm. Hence, they are essentially texture free and display
no deformation structure, as commonly observed in
extruded profiles.
From each of the billets, flat tensile specimens (with
cross section 3 mm 910 mm), round tensile specimens
(with diameter 6 mm), and 10 mm diameter round
compression specimens (with height 15 mm) were
machined. These were subsequently duly heat treated
to obtain the desired temper conditions prior to
mechanical testing. Details of the applied heat treatment
schedules are given in Table II.
The tensile testing of the flat and round tensile
specimens was carried out at a constant strain rate of 0.1
and 0.01 s
1
, respectively, using the standard testing
equipment available at NTNU/DMSE. Moreover, the
compression testing was performed in a Gleeble
machine at Hydro R & D Materials Technology at a
constant strain rate of 0.1 s
1
. The authors recognize
that these two strain rates are 100 to 1000 times faster
than those normally applied in conventional tensile
Fig. 4—Method used to calculate the geometric slip distance and the
corresponding volume fraction of nonshearable precipitates in Al-
Mg-Si alloys. In the model, these Orowan particles are characterized
by their radius r
o
>r
c
, where r
c
is the critical particle radius for dis-
location bypassing.
Table I. Chemical Composition of Alloys Used for Calibration and Validation of Work Hardening Model (Weight Percent)
Alloy Si Mg Cu Mn Fe Others Al Ref.
AA6082* 1.04 0.67 0.03 0.54 0.20 — balance 39
AA6063* 0.44 0.46 0.006 0.03 0.19 — balance 39
*Homogenized at 580 °C for 3 h. The applied heating and cooling rates were 200 °C and 350 °C/h, respectively.
Table II. Temper Designations and Applied Heat Treatment Schedules for Alloys Used to Calibrate and Validate the Work
Hardening Model
Alloy Temper Designation Supplementary Details Ref.
AA6082 W10 SHT
1
* followed by 10 min storage at room temperature 39
AA6082 T6 SHT
2
** followed by aging at 185 °C for 5 h 39
AA6082 T7 SHT
2
** followed by aging at 185 °C for 1 week 40
AA6063 W10 SHT
1
* followed by 10 min storage at room temperature 39
AA6063 T6x SHT
2
** followed by aging at 185 °C for 1 h 40
AA6063 T6 SHT
2
** followed by aging at 185 °C for 5 h 39
AA6063 T7 SHT
2
** followed by ageing at 185 °C for 1 week 40
*SHT
1
: solution heat treated at 540 °C for 15 min followed by water quenching.
**SHT
2
: solution heat treated at 530 °C for 15 min followed by water quenching.
METALLURGICAL AND MATERIALS TRANSACTIONS A

testing of metallic materials. However, this is permissible
in the present case, because previous investigations have
shown that the room-temperature flow stress of Al-Mg-
Si alloys is essentially independent of the applied strain
rate within the range from 0.00001 to 0.1 s
1
.
[29]
Further
experimental details are provided in References 39
and 40.
B. Calibration Procedure
Referring to Tables Iand II, the following combina-
tions of alloys and temper conditions are used to
calibrate the model, i.e., AA6082-W10 and AA6063-
W10, to determine the adjustable parameters k
1
and k
2
in the Voce equation and AA6082-T7 to determine qref
g;s
and eref
cin the unified work hardening model.
1. Adjustable parameters in Voce equation
In the as-quenched (W10) condition, only statistically
stored dislocations will contribute to the observed
work hardening behavior. Hence, the Voce equation
(i.e., Eq. [8]) can be used to determine the two adjustable
parameters k
1
and k
2
in the work hardening model.
Fine-tuning of the parameters to the experimental
tensile and compression test data for AA6082-W10
reproduced in Figure 5gives k1¼4108m
1
and
k
2
= 12.9.
The next step is to evaluate how variations in the
solute content influence the rate of dynamic recovery in
the materials, as determined by the k
2
parameter in the
Voce equation. Because k
1
is assumed to be constant for
a given family of alloys, the solute dependence of k
2
may
be evaluated from Eq. [9], using experimental data for
the saturation stress in the as-quenched (W10) condition
for AA6082 and AA6063, along with an appropriate
weight parameter to balance out the individual contri-
butions from the two major alloying elements Mg and Si
in solid solution. In the context of the work hardening
model developed, this is most conveniently done by first
invoking the relationship commonly employed to
describe the solute dependence of the flow stress and
then relating the saturation stress Dr
d,s
directly to the
equivalent magnesium content ^
CMg in the alloy via the
equation
[41,42]
Drd;s¼k3^
C3=4
Mg ½19
taking the weight parameter ^
CMg equal to:
[43]
^
CMg ¼CMg þ0:5Ceff
Si ½20
where C
Mg
is the mean concentration of magnesium in
the aluminum matrix, and Ceff
Si is the corresponding
silicon concentration following a correction for the
amount of Si being tied up as coarse a-Al
15
(FeMn)
3
Si
2
particles in the alloy.
[27]
Note that the individual values of C
Mg
and Ceff
Si during
heat treatment of Al-Mg-Si alloys can readily be
obtained from the precipitation model through their
inter-relationship with C. This makes the ^
CMg parameter
flexible and applicable to a wide range of temper
conditions, implying a change in the matrix solute level.
By substituting Eq. [19] into Eq. [9], the following
expression is obtained for the solute dependence of the
k
2
parameter:
k2¼k1aMGb
k3^
CMg

3=4½21
As shown in Figure 6, this equation can readily be
fitted to the experimental data provided for AA6082-
W10 and AA6063-W10, using the input value for k
3
listed in Table III.
2. Additional adjustable parameters in the work
hardening model
In the overaged (T7) temper condition, both statisti-
cally stored dislocations q
s
and geometrically necessary
dislocations q
g
contribute to the observed work hard-
ening behavior. Therefore, as a starting point, the
precipitation model is used to calculate the actual values
of the different microstructure parameters in the
AA6082-T7 reference material, i.e.,^
CMg ¼^
Cref
Mg (to
determine k2¼kref
2), kg;o¼kref
g;o, and fo¼fref
o. Then, the
Fig. 5—Calibration of Eq. [8] to experimental tensile and compres-
sion test data compiled for AA6082-W10. Fig. 6—Calibration of Eq. [21] to experimental compression test
data compiled for AA6082-W10 and AA6063-W10.
METALLURGICAL AND MATERIALS TRANSACTIONS A

saturation stress Drref
d;sis read from the experimental
work hardening curve presented in Figure 7and substi-
tuted into Eq. [16] to calculate the corresponding
dislocation density at saturation qref
g;s, taking:
qref
g;s¼Drref
d;s
aMGb
!
2
k1
kref
2

2
½22
Finally, the reference value for the critical strain eref
cin
Eq. [12] can be evaluated from the full work hardening
curve in Figure 7by fine-tuning of Eq. [15] to the
experimental data, giving eref
c¼0:05.
Table III contains a summary of all input data used in
the unified work hardening model.
C. Comparison between Predictions and Experiments
The remaining combinations of alloys and heat treat-
ment schedules, as defined in Tables Iand II, are used to
validate the unified work hardening model. Figure 8
shows four comparative plots of the net flow stress Dr
d
vs
the accumulated plastic strain e
p
, using the experimental
tensile and compression data compiled for AA6063. It is
evident from these graphs that the model is capable of
reproducing the observed work hardening behavior
following heat treatment to the different temper condi-
tions reasonably well. In particular, the overall agreement
between predictions and experiments is satisfactory for
the as-quenched (W10) and peak-aged (T6) materials. On
the other hand, the model does not fully reproduce the
actual work hardening curves for the underaged (T6x)
and overaged (T7) materials, which, in practice, are seen
to level off more slowly than predicted before entering
steady state or reaching saturation. Also the experimental
work hardening curve for AA6082-T6 is adequately
captured by the model, as shown by the plot in Figure 9.
Finally, Figure 10 shows a compilation of calculated
and measured flow curves for AA6082, where the yield
strength r
y
has been added linearly to Dr
d
in order to
obtain the resulting flow stress r, in agreement with
Eq. [5]. In practice, it is the actual value of r
y
that
determines the locus of the flow curve in stress-strain
space. It follows that the model correctly predicts the
yield strength of both the as-quenched (W10) and
overaged (T7) materials. In contrast, r
y
is underesti-
mated by nearly 50 MPa in the case of the peak-aged
(T6) material. Therefore, the predicted flow curve is seen
to lie systematically below the measured one within the
entire range in plastic strain examined, although its
shape is adequately captured by the work hardening
model, as shown previously in Figure 9. The reason for
this shortcoming of the yield strength model is not
known, but probably reflects the fact that it has not
previously been validated for billet materials.
D. Summary of Work Hardening Behavior of AA6063
and AA6082
In spite of these limitations, it is obvious that the
unified work hardening model is sufficiently relevant and
comprehensive to justify calculations of the full stress-
strain curves for AA6060 and AA6082 shown in
Figures 11 and 12, respectively, covering the different
temper conditions listed in Table II. Note that the
starting point of these hardening curves is obtained
using the yield strength model, which also relies on
outputs from the precipitation model, as illustrated
previously in Figure 1. In the past, this model has been
calibrated and benchmarked against experimental data
for extruded profiles of AA6063 and AA6082,
[27,28]
and
should thus yield the correct r
y
values for such materials
at the indicated temper conditions.
Table III. Summary of Input Parameters Used in Unified Work Hardening Model
Parameter Value Comments
a0.30 from Reference 10
b(m) 2.84 910
10
magnitude of Burgers vector
G(N/m
2
) 2.7 910
10
magnitude of shear modulus
M3.1 magnitude of Taylor factor
k
1
(m
1
)4108calibration of Eq. [8] to data in Fig. 5
k
3
(N/m
2
wt pct
3/4
)2108calibration of Eq. [21] to data in Fig. 6
kref
g;o(m) 4.1 910
7
calculated from precipitation model
fref
o0.0109 calculated from precipitation model
r
c
(m) 5 910
9
from References 27 and 28
qref
g;s(m
2
) 4.93 910
13
calculated from Eq. [22] with data from Fig. 7
eref
c0.05 calibration of Eq. [15] to data in Fig. 7
Fig. 7—Calibration of Eq. [16] to experimental tensile and compres-
sion test data compiled for AA6082-T7. The resulting value for qref
g;s
is calculated from Eq. [22].
METALLURGICAL AND MATERIALS TRANSACTIONS A

Fig. 8—Validation of unified work hardening model based on experimental tensile and compression test data compiled for AA6063: (a) as-quen-
ched (W10) condition, (b) underaged (T6x) condition, (c) peak-aged (T6) condition, and (d) overaged (T7) condition.
Fig. 9—Validation of unified work hardening model based on exper-
imental tensile and compression test data compiled for AA6082-T6.
Fig. 10—Summary of calculated and measured flow curves for
AA6082 following heat treatment to different temper conditions (i.e.,
W10, T6, and T7). In these plots, the yield strength r
y
is added line-
arly to Dr
d
in order to obtain the resulting flow stress r, in agree-
ment with Eq. [5].
METALLURGICAL AND MATERIALS TRANSACTIONS A

The curves presented in Figures 11 and 12 summarize
in an explicit manner the interplay between the two
main variables controlling the work hardening behavior
of commercial Al-Mg-Si alloys, i.e., the alloy composi-
tion and the applied heat treatment schedule. In the as-
quenched (W10) condition, the yield strength is low,
while the work hardening potential is large, owing to the
high solute level. Aging to the T6 temper implies that the
yield strength increases to its peak level at the same time
as the contribution from work hardening to the total
flow stress is correspondingly reduced due to solute
depletion. Finally, in the overaged (T7) temper condi-
tion, the yield strength drops again, mainly because of
particle coarsening, in combination with excessive
draining of solute from the aluminum matrix. On the
other hand, since these precipitates are all of the
Orowan type, storage of geometrically necessary dislo-
cations in their vicinity leads to a high initial work
hardening rate. However, this rapid increase in r
suddenly diminishes when the accumulation of disloca-
tion loops around the precipitates ceases and q
g,s
is
reached at some critical plastic strain e
p
=e
c
(Figure 3),
which causes the flow stress to saturate immediately
thereafter.
IV. APPLICATIONS OF UNIFIED WORK
HARDENING MODEL
In the following, the versatility of the combined
precipitation, yield strength, and work hardening mod-
els will be illustrated using three different examples, all
being relevant to manufacturing of aluminum compo-
nents for automotive applications.
A. Construction of Dislocation Mechanism Maps
Figures 13(a) and (b) show how the saturation values
for the three dislocation densities q
s,s
,q
g,s
, and q
t,s
vary
with time following artificial aging at 185 °C for
AA6063 and AA6082, respectively. The different temper
conditions listed in Table II are indicated by arrows
along the abscissas, whereas the yield stress r
y
and the
corresponding saturation value for the flow stress r
sat
can be read from the ordinate on the right-hand side of
the diagrams, taking:
rsat ¼ryþaMGbffiffiffiffiffiffiffi
qt;s
p½23
It is evident from these mechanism maps that the
contribution from the statistically stored dislocations
dominates during the initial and final parts of the heat
Fig. 11—Predicted stress-strain curves for AA6063 following heat
treatment to different temper conditions.
Fig. 12—Predicted stress-strain curves for AA6082 following heat
treatment to different temper conditions.
Fig. 13—Mechanism maps showing how the saturation values for
the three dislocation densities, q
s,s
,q
g,s
, and q
t,s
, as well as the yield
stress r
y
and the saturation value for the flow stress r
sat
, vary with
time following artificial aging at 185 °C: (a) AA6063 and (b)
AA6082.
METALLURGICAL AND MATERIALS TRANSACTIONS A

treatment process when the particle-dislocation interac-
tion is weak. In contrast, the geometrically necessary
dislocations contribute most to the work hardening
behavior in the peak-aged and partly overaged temper
conditions immediately following the appearance of the
Orowan particles, which particularly in the case of
AA6063 is accompanied by a rapid increase in r
sat
. The
situation persists until the geometric slip distance k
g,o
again becomes larger than the corresponding slip
distance for statistical storage of dislocations k
s
in the
material because of particle coarsening, which eventu-
ally makes Orowan looping insignificant compared to
the other work hardening mechanism.
B. Plastic Instability in Tension
The full stress-strain curves shown previously in
Figures 11 and 12 provide a basis for evaluating the
point of plastic instability in tension, using the well-
established criterion for the onset of necking at maxi-
mum load:
[44]
dr
dep¼r½24
From this relationship, the true uniform plastic strain e
u
can be calculated for different combinations of alloy
compositions and heat treatment schedules.
The true uniform plastic strain is a parameter that is
relevant to many cold forming operations involving
tensile loading. Figure 14 shows how e
u
varies with
aging time at 185 °C for AA6063 and AA6082. Again,
the different temper conditions listed in Table II are
indicated by arrows along the abscissa. As expected, the
true uniform plastic strain is seen to drop following
artificial aging due to solute depletion, which reduces the
work hardening potential of the alloys and thus the
intrinsic resistance to necking during tensile testing.
Also the dependence of e
u
on the alloy composition is
reflected in these plots, the solute-rich AA6082 material
being generally the most ductile one in the as-quenched
(W10) condition because of its superior work hardening
behavior.
C. Localized Deformation within the HAZ of Fusion
Welds
Finally, an example is provided where the combined
precipitation, yield strength, and work hardening mod-
els are linked to the FE code WELDSIM
[45–47]
to
evaluate the stress-strain response of distinct regions
within the heat-affected zone (HAZ) following gas metal
arc welding (GMAW) of an AA6060-T6 tube material.
Further details are given in Reference 30.
The predicted peak temperature distribution during
welding and the resulting hardness profile across the HAZ
are shown in Figures 15(a) and (b), respectively. Based on
the input from WELDSIM, the full stress-strain curves
have been calculated for the three positions labeled A, B,
and C in Figure 15(a). These curves are shown in
Figure 15(c). It follows that each region yields a unique
stress-strain response, where position B corresponds to
the softest part of the HAZ. This results in localized
deformation and eventually in mechanical failure of the
aluminum component during loading, which is devastat-
ing for the resulting crash performance.
[30]
V. CONCLUSIONS
The basic conclusions that can be drawn from this
investigation are as follows.
In general, a two-internal-variable model is required to
capture the complex interactions between solute atoms,
precipitates, and dislocations being responsible for the
observed work hardening behavior of commercial
Al-Mg-Si alloys at room temperature. These internal
variables should, in turn, be linked to an underlying
precipitation model to provide a basis for calculating the
different solute and precipitate parameters that control
the resulting stress-strain response following heat
treatment.
It is shown that an analytical formulation of the work
hardening problem is possible if the total dislocation
density is split into statistically stored and geometrically
necessary dislocations, respectively, as originally pro-
posed by Ashby.
[12]
This, in turn, allows a separate
treatment of each group of dislocations, using the latter
parameters as the independent variables of the system.
By converting the current values of the dislocation
densities into an equivalent room-temperature flow
stress using classic dislocation theory, the change in
the work hardening behavior as a result of heat
treatment and welding can be evaluated for different
alloy compositions. This is possible because the work
hardening model, in a calibrated form, relies solely on
outputs from the previously developed precipitation and
yield strength models for the same class of alloys.
In addition to the solute content, which determines
the rate of dynamic recovery, the two other microstruc-
ture parameters that control the room-temperature
work hardening behavior of commercial Al-Mg-Si
alloys are the geometric slip distance and the corre-
sponding volume fraction of nonshearable Orowan
particles. Both parameters can be extracted from the
predicted particle size distribution.
Fig. 14—Plots showing how the true uniform plastic strain e
u
varies
with time following artificial aging at 185 °C for AA6063 and
AA6082. The calculations are done using the instability criterion
defined by Eq. [24].
METALLURGICAL AND MATERIALS TRANSACTIONS A

Finally, it is concluded that the combined precipita-
tion, yield strength, and work hardening model outlined
previously provides a powerful tool for predicting
materials properties and response behavior that also
can be exploited in engineering design and used as input
to other, more complicated mechanical models. The
examples given range from construction of dislocation
mechanism maps via prediction of plastic instability in
tension to calculations of local stress-strain curves
within the HAZ of fusion welds, all being relevant to
manufacturing of aluminum components for automo-
tive applications.
ACKNOWLEDGMENTS
The authors acknowledge the financial support pro-
vided by the Norwegian Research Council and Hydro
Aluminium through SIMLab, the Centre for Research-
Based Innovation, at the Norwegian University of Sci-
ence and Technology. Moreover, they are grateful to
Dr. Øyvind Ryen for providing the experimental ten-
sile and compression test data used to calibrate and
validate the work hardening model.
APPENDIX
Symbols and units
A
0
parameter related to the energy barrier for
nucleation (J/mol)
bmagnitude of the Burgers vector (m)
Cmean solute concentration in matrix (wt pct)
C
0
nominal solute concentration in matrix (wt pct)
C
e
equilibrium solute concentration at the particle/
matrix interface (wt pct)
CSi
eff effective silicon content in alloy (wt pct)
C
i
solute concentration at the particle/matrix
interface (wt pct)
C
Mg
solute concentration of magnesium in matrix (wt
pct)
^
CMg equivalent magnesium concentration (wt pct)
^
Cref
Mg value of ^
CMg in reference alloy (wt pct)
C
p
concentration of alloying element inside the
particle (wt pct)
Ddiffusion coefficient (m
2
/s)
fparticle volume fraction
f
o
volume fraction of nonshearable Orowan
particles
fref
ovalue of f
o
in reference alloy
Gshear modulus (N/m
2
)
jnucleation rate (#/m
3
s)
j
0
pre-exponential term in expression for j(#/m
3
s)
k
1
parameter related to statistical storage of
dislocations (m
1
)
k
2
parameter related to dynamic recovery of
dislocations
kref
2value of k
2
in reference alloy
k
3
parameter determining the solute dependence of
k
2
(N/m
2
wt pct
3/4
)
‘mean planar particle spacing along the bending
dislocation (m)
MTaylor factor
N
i
number of particles per unit volume within the
size class r
i
(#/m
3
)
Q
d
activation energy for diffusion (J/mol)
Runiversal gas constant (8.314 J/Kmol)
rparticle radius (m)
Fig. 15—Outputs from the coupled numerical simulations of the
welding case referred to in the text: (a) predicted peak temperature
T
p
distribution during GMAW; (b) resulting hardness distribution
across the HAZ with the scale for T
p
being indicated along the up-
per abscissa; and (c) calculated stress-strain curves for the three posi-
tions labeled A, B, and C in (a).
METALLURGICAL AND MATERIALS TRANSACTIONS A

r
i
particle radius within size class i(m)
r
c
critical particle radius for the transition from
shearing to bypassing (m)
r
o
particle radius of nonshearable Orowan particles
(m)
ttime (s)
Ttemperature (K or °C)
T
p
peak temperature (K or °C)
aconstant in expression for Dr
d
Dr
d
net contribution from dislocation hardening to
flow stress (N/m
2
)
Dr
d,s
value of Dr
d
at saturation (N/m
2
)
Drref
d;svalue of Dr
d,s
in reference alloy (N/m
2
)
Dplate thickness (m)
etensile strain
e
c
critical strain at which q
g
saturates
eref
cvalue of e
c
in reference alloy
e
p
plastic tensile strain
e
u
true uniform plastic strain
cshear strain
c
ref
value of cin reference alloy
k
g
geometric slip distance (m)
k
g,o
geometric slip distance of Orowan particles (m)
kref
g;ovalue of k
g,o
in reference alloy (m)
k
s
slip distance for statistical storage of dislocations
(m)
q
g
number density of geometrically necessary
dislocations (m
2
)
qref
gvalue of q
g
in reference alloy (m
2
)
q
g,s
value of q
g
at saturation (m
2
)
qref
g;svalue of q
g,s
in reference alloy (m
2
)
q
s
number density of the statistically stored
dislocations (m
2
)
q
s,s
value of q
s
at saturation (m
2
)
q
t
total dislocation density (m
2
)
q
t,s
value of q
t
at saturation (m
2
)
rflow stress (N/m
2
)
r
i
intrinsic yield strength of pure aluminum (N/m
2
)
r
p
contribution from hardening precipitates to the
overall macroscopic yield strength (N/m
2
)
r
ss
contribution from alloying elements in solid
solution to the overall macroscopic yield
strength (N/m
2
)
r
sat
value of rat saturation (N/m
2
)
r
y
overall macroscopic yield strength (N/m
2
)
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