# Predicting the effect of strain path on the strain aging behaviour of Ultra-
low carbon steel

W. A. Melo  C. W. Sinclair<EMAIL_ADDRESS>Department of Materials
Engineering, The University of British Columbia, 309-6350 Stores Road,
Vancouver, Canada

###### Abstract

A recently presented model is adapted to predict the effects of strain path
change on the strain aging (bake hardening) behaviour of an ultra low carbon
(ULC) steel. Samples pre-deformed by rolling were aged and then tested in
uniaxial tension at 0, 45 and 90∘ degrees to the prior rolling direction. The
results show that aging changes not only the yield strength of the material
but also the work hardening rate. The increase in yield strength is
interpreted to be dominated by a reduction in mobile dislocation density, this
reflecting the classic ideas of Hahn and others. The change in work hardening
behaviour is treated as arising from the different role played by solute
‘loaded’ dislocations compared to dislocations generated by straining after
aging.

###### keywords:

bake hardening, strain aging, steel, work hardening, yield strength, modelling

##  1 Introduction

Ultra-low carbon (ULC) steels rely primarily on two phenomena to achieve
strength in formed parts; work hardening and strain aging. These two features
work in tandem in the bake hardening process; steel is first deformed, to work
harden it, followed by annealing to allow for a further increase in strength
due to interstitial solute re-arrangement and segregation to dislocations
(strain aging or bake hardening) (see e.g. [1, 2]). Laboratory
characterization of strain aging in ULC steels is conventionally carried out
by performing tensile pre-straining ($\sim$2-5% tensile strain, consistent
with the strain imposed during stamping operations) then aging at 150-200∘C
for 20-30 minutes to simulate the industrial paint bake cycle [1]. This aged
material is then tested again in tension, the orientation of the tensile axis
being kept the same between tests. The typical result of this process,
highlighted in Figure 1a, is an increase in the yield strength post strain
aging, with the appearance of an upper yield point and Lüders plateau.
Commercially, bake hardening (BH) is quantified as the increases in yield
stress following the aging treatment. This definition uses the difference
between the flow stress at the end of pre-deformation and the upper or lower
yield stress measured after aging. The aging time-temperature dependence of BH
has been studied in detail with a number of analytical models that allow for a
phenomenological description of this behaviour, see e.g. [3, 1, 4, 5]. Despite
this convention for the measurement of BH, the strain path experienced by
steel components during fabrication and service are usually neither uniaxial
tension nor performed in the same direction pre and post aging. For example,
in sheet metal forming for automotive applications the pre-deformation
typically ranges from plane strain to equibiaxial tension, while the
deformation of interest post aging, e.g. ‘denting’, is also far from simple
uniaxial tension [1, 6]. The effect of strain path change following strain
aging has been most heavily studied under conditions where the strain path is
reversed between the pre-deformation and the deformation post aging. Without
aging, such tests (e.g. tension-compression or torsion-reverse torsion) are
already complicated by the Bauschinger effect [7]. For pre-strain levels
commensurate with commercial bake hardening (a few percent), anisotropy of
mechanical response on strain path change is most often (though not always,
see e.g. [8]) considered to arise due to low misorientation dislocation cells
formed in the first stage of deformation [9, 10, 11, 12]. These cells can act
as strong barriers to the operation of (new) slip systems on reloading, though
typically the material evolves quickly back towards the work hardening
response it would have had if it had been deformed monotonically [9, 11]. In
work on steel without strain aging, Wilson and Bate [9] showed that yielding
upon load path change was accompanied (microscopically) by the inhomogeneous
operation of slip systems in some grains first, leading to localized slip
bands. Continued deformation then led to plasticity spreading through the
sample.

When strain aging is performed between deformation steps where a strain path
change is involved, the most common observation, for a range of steels, has
been that strain aging acts to reduce, though never totally eliminate, the
Bauschinger effect [13, 14, 15, 16]. This means that, unlike tension-tension
tests where the BH effect is characterized by an upper and lower yield point
following aging, in strain path reversal tests the stress-strain curve remains
smooth and the deformation remains uniform within the gauge section of the
sample. Examples of non-reversed strain path changes with strain aging (e.g.
tension-aging-tension at an angle to the original tensile direction) are rare
in the literature and the results less clear concerning the impact on the
mechanical response. For example, Jun et al. [17] reported that the yield
strength increases in tension-tension tests, regardless of the angle between
the first and second directions of loading, while Hiwatashi et al. [18]
reported both increase and decrease in yield strength.

What is consistent among all studies where strain aging is conducted with a
change in strain path is that the strong yield point phenomena characteristic
of strain aged samples subjected to monotonic tension-tension tests is not
present. However, aging conducted with a strain path change does result in a
significant change in the work hardening behaviour. This was noted by Ballarin
et al. [1, 6] who developed a phenomenological model to predict not only the
yielding but also the work hardening response of samples subjected to strain
aging and non-uniaxial, non-monotonic strain paths. The model, while able to
reproduce the complex features of the strain aging response observed under a
variety of strain path changes, relies on a large number of empirical
parameters which need to be adjusted for each condition, this limiting the
model’s predictive capabilities. The work presented below seeks to build from
the work of Ballarin et al. [6] by proposing an alternative model for
predicting the strain aging response of ULC steel samples subjected to a
strain path change. The physically-based model used here is adapted from
recent work which itself is strongly motivated by the classic work of Hahn
[19]. Experiments are presented both for conventional tension-tension strain
aging tests as well as for tests where samples were pre-deformed in cold
rolling, aged, then tested in tension. The strengths and weaknesses of this
approach are highlighted as are areas for potential future work.

##  2 Materials and Experimental Method

For this study a 1 mm thick ULC steel sheet (composition in Table 1) was
supplied by ArcelorMittal Dofasco. The material, received in the as-cold
rolled state, was annealed in a box furnace in air at 800∘C for 7 min, the
temperature of the samples monitored by a thermocouple spot welded to them.
Annealing resulted in a fully recrystallized microstructure with an average
equal area grain diameter of 14 $\mu$m.

C | N | Si | P | Mn | Nb | Ti | S | Al (tot) | Cr | Ni | Cu | Other
---|---|---|---|---|---|---|---|---|---|---|---|---
0.0024 | 0.0027 | 0.011 | 0.039 | 0.358 | 0.0048 | 0.012 | 0.0084 | 0.042 | 0.043 | 0.023 | 0.037 | 0.006

Table 1: Nominal composition of studied ULC steel. Balance is iron.

In order to study strain aging on this material, two forms of pre-deformation
were used; uniaxial tensile tests were performed on one set of samples, while
another set of samples were pre-deformed by performing laboratory cold
rolling. In both cases, following aging, the second deformation (to measure
the strain aging effect) was performed in uniaxial tension. For all tensile
tests the same dog-bone shaped tensile samples were used (gauge length of 40
mm, gauge width of 6 mm). These were tested at room temperature in an Instron
load frame with a fixed cross-head speed of 0.21 mm/s, corresponding to a
nominal strain rate of approximately $6\times 10^{-2}$ s-1. Strains were
measured using a clip-on extensometer. For the tensile tests performed to pre-
strain the material, the tests were stopped at 10% total strain, this being
just beyond the end of the Lüder’s plateau in the stress strain curves.

For samples pre-deformed by rolling, the deformation was conducted using an
unlubricated laboratory rolling mill. The rolling direction was kept the same
as that used in the commercial production of the sheet. All sheets were cold
rolled in 4-5 passes to a reduction in thickness of 9 $\pm$ 1%. The reduction
in thickness was measured using a digital micrometer at several locations in
the as-rolled sheet. From these rolled sheets, tensile samples were cut
parallel (0∘) to the sheet’s rolling direction (RD), transverse (90∘) to RD,
and at 45∘ to RD. Tensile samples cut at 0 and 90∘ to RD were processed in two
sheets rolled at the same time, meaning that the same reduction per pass and
total reduction were achieved. Samples cut at 45∘ to RD were taken from a
sheet rolled separately.

Following pre-deformation in tension or in rolling, samples were aged in a
stirred silicone oil bath. The bath temperature was monitored using a K-type
thermocouple submerged next to the samples. For the samples pre-deformed in
tension, the aging treatment was performed within 10 minutes of the end of the
pre-deformation step. For samples pre-deformed in rolling, a longer delay
between deformation and aging was required owing to the need to machine
tensile samples. In this case, aging was limited to long times/high
temperatures so as to minimize the effects of further aging at room
temperature.

Two-dimensional digital image correlation (DIC) was performed on selected
samples during tensile testing to clarify the homogeneity of strain in the
gauge section. For DIC measurements a single camera was used with the LAVISION
DIC system for image capture and subsequent processing. A random speckle
pattern was deposited on the surface of these samples using black paint and an
airbrush. During DIC image acquisition the measured load from the load cell
was recorded on each image. This allowed for a direct correlation between the
DIC measurements and the stress-strain response. As a validation of the
technique, the average axial strain in the gauge section of the sample was
used, along with the recorded load, to calculate the Young’s modulus of
samples during unloading. The results were found to always fall between
204-208 GPa.

Finally, to assist with the prediction of the stress-strain response under
conditions where non-uniform strains were observed during tensile testing (due
to Lüders bands), simulations of tensile tests were performed using ABAQUS CAE
version 6.14 FEA software [20]. The full tensile sample geometry was used for
these simulations, with linear 8-node brick elements (C3D8R). Elements within
the gauge section were 0.245 mm wide, 0.3 mm long and 1 mm in thick (1 element
in the thickness of the sample). An encastre boundary condition (no
displacement or rotation) was imposed on the nodes in the lower grip section,
while nodes on the upper grip surface were subjected to a displacement
parallel to the tensile axis, with no displacements/rotations allowed in any
other directions. The material in the gauge section was subjected to no
mechanical constraint. To allow for the controlled nucleation of a Lüders
band, a very small geometric imperfection (similar to a through thickness
v-shaped notch) 90 $\mu$m deep and 20 $\mu$m wide was introduced at the
junction between the top-left fillet and gauge section of the sample.

##  3 Strain Aging Kinetics with a Monotonic Strain Path

Strain aging tests were first performed under conventional tension-aging-
tension conditions to assess the aging kinetics under the conditions used
here. Aging was conducted at 50, 100 and 150∘C for times ranging from 1 min to
220 min. Figure 1a shows the stress-strain curve for an unaged, or ‘as-
annealed’, tensile sample as well as a sample aged for 10 min at 150∘C, this
illustrating the typical changes in stress-strain response following aging.
Also shown in the inset is the DIC measured propagation of the Lüders bands in
the as-annealed sample, the bands originating at the top and bottom of the
sample’s gauge section. Uniform deformation commences once these bands fill
the gauge section of the sample. Figure 1b summarizes the strain aging
kinetics for all conditions studied, using the change in the upper yield
stress ($\sigma_{U}$) as a measure of the strain aging/BH effect. The solid
lines in these plots represent a fit of the data to the classic Louat-
Cottrell-Bilby model [21, 22],

$\frac{\Delta\sigma_{u}}{\Delta\sigma_{max}}=1-\exp{\left(-\left(\frac{t}{t^{*}}\right)^{2/3}\right)}$
(1)

Using the data here it was found that a good prediction could be obtained
using $\Delta\sigma_{max}=55$ MPa and $t^{*}=2.34\times 10^{-16}/D$ where
$D=D_{0}\exp{\left(-\frac{Q}{RT}\right)}$ is the the diffusivity of carbon in
$\alpha$-Fe, with $D_{0}=1.2\times 10^{-4}$ m2/min and $Q=85$ kJ/mol, as used
in [23]. The results shown in Figure 1 agree well with previous results
published in the literature for similar grades of steel, tested under similar
circumstances. While others have pointed to the possibility of additional
hardening coming from the precipitation of carbides at higher
temperatures/longer aging times, here the data seems to be adequately captured
by the single set of kinetics represented by equation 1.

Figure 1: a) Stress-strain curve of as-annealed material (blue) and strain
aged material (10 min at 150∘C after 10% tensile pre-deformation). Also shown
for illustration are DIC images taken while the samples were undergoing Lüders
band propagation for the as-annealed material. b) strain aging kinetics for
different times and temperatures following a 10% pre-deformation in tension.
The solid lines show the best fits of equation 1.

##  4 Strain Aging with a Change in Strain Path

Based on the above results, an aging condition of 10 min at 150∘C was selected
for the rolled samples, as it represented the maximum BH effect observed in
figure 1. As noted above, this ‘peak’ strain aged condition was selected so as
to minimize the affect of the samples needing to be left at room temperature
for several hours prior to aging so that tensile samples could be machined.

Like with the results shown in figure 1, one would like to compare the
mechanical (tensile) response after aging with the mechanical response of the
material in the as-recrystallized state. The challenge, in these experiments,
arises from the ambiguity associated with defining an appropriate measure of
equivalent plastic tensile strain to associate with the rolling pre-
deformation. Assuming the material obeys a Von Mises (J2) flow theory, one can
estimate the equivalent plastic strain based on the thickness reduction
($\epsilon_{t}$) assuming monotonic plane strain deformation,

$\bar{\epsilon}=\frac{2}{\sqrt{3}}\epsilon_{t}$ (2)

This, however, should be considered as a lower bound estimate as it ignores
the complex deformation path experienced as the material passes through the
roll gap, including the effects of friction (see e.g. [24, 25]).

Additional data which can help reduce this ambiguity comes if we compare the
work-hardening/flow stress relationship (Kocks-Mecking plot [26]) for samples
with and without pre-deformation in rolling. It is expected that the change in
strain path may introduce a transient in the work hardening response [27, 28,
29], but that beyond this transient the work hardening-flow stress response,
dominated by the dislocation density in this case, will reflect the underlying
hardening response of the sample [30, 26]. Figure 2a-c shows the engineering
stress-strain response of the as-rolled and rolled and aged samples tested at
0∘, 45∘ and 90∘ to the prior rolling direction. In each case, the stress-
strain curves are plotted to the onset of necking (plotted to the ultimate
tensile strength). As can be seen, in the case of the sample tested in the as-
rolled condition at 90∘ to the tensile axis, necking commences shortly after
yielding. Figure 2d and e show the same data, but in this case as Kocks-
Mecking plots where the work hardening behaviour of the starting as-annealed
material (post Lüdering) is also shown. Here the work hardening rates were
computed from the true-stress, true strain data smoothed by interpolating
using a B-spline with 5 knots and degree of 3 (as implimented in the splev
function in scipy [31]). As conventional for cubic metals, the Kocks-Mecking
plot for the as-annealed material shows a linear decrease in work hardening
with increasing flow stress [30, 26], the thin black line illustrates this for
the three cases shown in figure 2d. The pre-deformed sample tested at 0∘ to RD
shows a similar response and a work hardening behaviour that nearly overlaps
(after an initial transient) with that of the as-recrystallized material. This
coincides with previous work which has shown that for steels, pre-deformed in
rolling, small pre-strains in rolling followed by tensile testing parallel to
RD gives the smallest strain-path change effect to the resulting stress-strain
curve [9]. In contrast, the rolled sample tested at 45∘ to RD shows a more
significant transient in the work hardening response. The sample tested at 90∘
to RD could not be analyzed owing to the early onset of localization.

Figure 2: a)-c) Stress-strain curves for samples after pre-deformation and
pre-deformation and aging for samples rolled then tensile tested at 0∘, 45∘
and 90∘ to the rolling direction. The aged samples were treated at 150∘C for
10 min. d) work hardening versus flow stress plot comparing the work hardening
response (post Lüdering) to the work hardening response of the rolled and
tensile tested samples e) as (d) but for samples that were aged after rolling.

Figure 2e shows that following aging, not only does the yield strength of the
material change but so too does the work hardening behaviour. In the case of
all samples rolled, aged and tested at 0, 45 and 90∘ to RD, the initial work
hardening increases as does the rate of drop of the work hardening rate with
stress. This observation would be unobtainable from a simple model that
considers only changes in the yield strength of the material on aging (cf.
equation 1).

Following from the results in figure 2, one can estimate an effective tensile
strain resulting from rolling, this allowing us to adjust the initial strain
used to plot the stress-strain curves in figure 2a-c when comparing with the
stress-strain curves of the as-annealed material. From figure 2 we see that
the work hardening behaviour of the samples rolled, then tested at 0 and 45∘
to RD follow a similar work hardening response to that of the as-annealed
material, except for the transient observed in the behaviour of the sample
tested at 45∘. Using $\bar{\epsilon}>0.1$ as a lower bound for estimating the
equivalent strain introduced by rolling, we find that a good agreement between
the stress-strain response of the as-annealed and rolled and tested samples
can be obtained using $\bar{\epsilon}=0.14$ for the samples tested at 0∘ to RD
while the samples tested at 45∘ match well to a continuation of the stress-
strain response of the as-annealed material with $\bar{\epsilon}=0.16$ (figure
3). Of course, it is impossible to compare the stress-strain response of the
as-rolled material tested at 90∘ to RD with the as-annealed material at higher
strains as the ultimate tensile strength occurred just after yielding. In this
case, however, since the samples taken at 0 and 90∘ to RD were cut from the
same rolled sheet we have applied the same $\bar{\epsilon}=0.14$ to both
cases.

Figure 3: a)-c) Stress-strain curves for samples after pre-deformation and
pre-deformation and aging for samples rolled then tensile tested at 0∘, 45∘
and 90∘ to the rolling direction. Also shown are the stress-strain curves for
the as annealed material. The stress-strain curves for the rolled and
rolled+aged samples have been translated in strain along the x-axis by
$\bar{\epsilon}=0.14$ for the samples tested at 0∘ and 90∘ to RD while the
results for samples tested at 45∘ to RD were translated by
$\bar{\epsilon}=0.16$. The light line in the background (labeled ‘Kocks-
Mecking’ is the Kocks-Mecking model (equations 3, 5) fit to the post-Lüdering
portion of the as-annealed stress-strain curve.

A final point regards the onset of yielding following rolling and aging. While
the classic response following strain aging is the one shown in figure 1a,
where after strain aging the material exhibits an upper and lower yield point
accompanying a yield point elongation, the stress-strain curves shown in
figure 2 appear quite different. The samples rolled, aged then tested at 0 and
45∘ to RD both show some evidence of a low, transient hardening behaviour on
yielding but this is not accompanied by any clear evidence of yield point
elongation. In the case of the sample rolled, aged and tested at 90∘ to RD,
yielding gives way to a smooth work hardening response with no clear evidence
of any transient. To confirm that yielding occurred uniformly along the gauge
length of these samples, DIC observations were made (figure 4), these being
compared to DIC measurements made on the gauge section of a sample pre-
strained in tension, aged and then tested again in tension (cf. figure 1b). In
this case we can see that the strain for the rolled and aged samples appears
relatively uniform in the gauge section of the samples while the formation of
Lüders bands in the tensile pre-deformed and aged samples is clearly seen by
the central portion of the sample having no measurable plastic strain. This
lack of Lüders band formation in the rolled and aged samples greatly
simplifies the analysis that will be presented next.

Figure 4: Digitial image correlation measured tensile strain (parallel to the
tensile axis) measured along the centre line of three tensile samples, the
macroscopic strain for each curve is shown in the legends. a) Measurement made
on a sample pre-deformed in tension, aged at 150∘C for 10 min then tested
again in tension (same condition as the sample shown in 1a). One can clearly
see the propegation of Lüders bands from both ends of the tensile sample
toward the centre. b) Tensile samples tested after rolling and aging at 0∘ and
c) 45∘ to RD. While both the stress strain curves in these conditions showed
some transient work hardening response on yielding, there is no evidence of
strain localization suggesting Lüders band formation or propegation.

##  5 Predicting The Effect of Strain Aging Following a Strain Path Change

As noted above, the classic approach to modelling the impact of strain aging
on the mechanical response of ULC steel considers only the yield strength
evolution. Clearly this is insufficient to explain the changes shown in figure
2 where both yield strength and work hardening rate evolve. A less used
approach for explaining the effects of strain aging on steels was developed
originally by Han [19]. In this approach, dislocations are considered to be
‘locked’ by interstitial solute but that it is dislocation multiplication that
controls the yielding behaviour, not dislocation ‘unlocking’. One can take
this approach a step further and consider the role that ‘locked’ or solute
‘loaded’ dislocations may have on the forest strengthening at higher strains.
This approach has recently been developed into a simplified two-state variable
model (building from the original work of Kubin and Estrin [32]) and applied
to Al-Mg alloys by Medrano [33]. Here we test this approach to predict both
the yield strength and work hardening response of the materials described
above. Rather than provide a detailed description of the model, we will focus
on providing a basic overview, readers interested in a more detailed
description are directed to [33].

The flow stress of the ULC steel is considered to be a function of the forest
dislocation density ($\rho_{f}$) and the imposed strain rate
($\dot{\epsilon}$) according to,

$\sigma=\sigma_{0}+M\alpha\mu
b\sqrt{\rho_{f}}\left(\frac{\dot{\epsilon}}{\dot{\epsilon}_{0}}\right)^{m}$
(3)

In equation 3, $\sigma_{0}$ is taken to be the yield strength of the as-
annealed material without the impact of strain aging, $M$ is the Taylor
factor, $\alpha$ is the Saada constant. As-annealed samples tested at 0, 45
and 90∘ to RD were seen to have different (upper) yield stresses but the
effect was small compared to those we hope to capture following aging, thus in
this case a single value $M=3$ was used for all samples and no intrinsic
anisotropy due to texture has been accounted for.

The reference strain rate, $\dot{\epsilon}_{0}$, is assumed proportional to
the mobile dislocation density $\dot{\epsilon}_{0}=\beta^{*}\rho_{m}$ [32]. In
[33] a simple approximation is taken for the evolution of the mobile
dislocation density, its value evolving towards a saturation value
$\rho_{m}^{sat}$ which is taken as a parameter in the model. This evolution is
controlled by a parameter $\eta$ as,

$\frac{d\tilde{\rho}_{m}}{d\epsilon}=\eta\left(1-\tilde{\rho}_{m}\right)$ (4)

where $\tilde{\rho}_{m}=\rho_{m}/\rho_{m}^{sat}$.

The evolution of the forest dislocation density, in the absence of aging, is
taken to obey the classic Kocks-Mecking phenomenological evolution equation
[30],

$\frac{d\rho_{f}}{d\epsilon}=k_{1}\sqrt{\rho_{f}}-k_{2}\rho_{f}$ (5)

With strain aging, however, one needs to consider the role that solute
‘loaded’ dislocations have on this equation for dislocation storage. In this
case, we additionally separate forest dislocations into two types, those which
have been ‘loaded’ with solute (density $\rho_{f}^{*}$) and those that are not
($\rho_{f}$). As in [33] we allow for the fact that both types of forest
dislocation will impact on the storage of additional forest dislocations
adding an additional term to the above evolution equation,

$\frac{d\rho_{f}}{d\epsilon}=k_{3}\sqrt{\rho_{f}^{*}}+k_{1}\sqrt{\rho_{f}}-k_{2}\rho_{f}$
(6)

In [33] an additional storage term due to mobile dislocations was included in
equation 6, consistent with [32], but it has been found here that this term
did not significantly contribute to improving the prediction (i.e. its pre-
factor was small) and so it has not been included so as to reduce the number
of parameters in the model.

Taking the above points into consideration, a single test on a solution
treated sample, allows one to establish values for $\sigma_{0}$, $k_{1}$ and
$k_{2}$, the values found for the alloy studied here being given in Table 2.
As in [33] the density of ‘loaded’ dislocations is allowed to be reduced by
deformation, this obeying the simple law,

$\frac{d\rho_{f}^{*}}{d\epsilon}=-k_{4}\rho_{f}^{*}\left[1-\exp{\left(\sqrt{\frac{\rho_{f}^{*}}{\rho_{f}}}\right)}\right]$
(7)

The results of this are a model which requires the evaluation of $\sigma_{0}$,
$k_{1}$, $k_{2}$, $k_{3}$, $k_{4}$, $m$, $\eta$, $\rho_{m}^{sat}$ and
$\rho_{f}^{*}$ at the end of strain aging in order to predict the stress-
strain response of a sample after deformation and aging. While the number of
parameters seems large we will show here that several of these must be held
constant over all conditions considered and that others are relatively
insensitive to model predictions.

We can first obtain estimates for $\sigma_{0},k_{1}$ and $k_{2}$ by
considering the behaviour of the as-annealed material. Specifically we assume
a condition where $\rho_{m}\approx\rho_{m}^{sat}$ and $\rho_{f}^{*}=0$. This
implies that rather than deal with fitting to the full stress-strain response
in figure 1a, which would require a spatially resolved model for the tensile
sample to include the inhomogeneous strain and strain rate at the Lüders band
front, here we fit our response to the post-Lüders portion of the stress-
strain curve. While the local strain rate (and therefore the evolved
dislocation density) within the portion of the stress-strain curve exhibiting
Lüdering will not be the same as that obtained from the same deformation
performed uniformly, we consider this for now a second order effect, something
we will return to at the end of the discussion. The values for these three
parameters were found from a best fit to the as-annealed work hardening curve
shown in figure 2 and are reported in table 2. The resulting smooth stress-
strain response is also shown in figure 2 (thin black line in d and e) and
figure 3 (as the light grey lines in the background of each figure).

With these parameters established, it is left to find
$k_{3},k_{4},\eta,\rho_{m}^{sat}$ as well as the initial values (after rolling
and/or rolling and aging) of $\rho_{f}$, $\rho_{f}^{*}$ and
$\tilde{\rho}_{m}$. The first four of these parameters are taken to be
constant regardless of the testing conditions (i.e. same values are used for
the as-rolled and the rolled and aged samples). In contrast, the values of the
initial dislocation densities ($\rho_{f}$, $\rho_{f}^{*}$ and $\rho_{m}$) will
be established for each of the test conditions as described below. As in [33],
the model is relatively insensitive to the value of $\eta$ and
$\rho_{m}^{sat}$, thus we have simply fixed these values to ones similar to
those used in [33]. These parameters mainly affect the initial yielding
response as the mobile dislocation density quickly approaches $\rho_{m}^{sat}$
on deformation. The values of $k_{3}$ and $k_{4}$ are particularly important
for predicting the initially high work hardening rate seen in figure 2e and
its rapid drop with straining. Eventually, as $\rho_{f}^{*}$ drops the work
hardening-flow stress relationship predicted by the above equations are
expected to asymptotically rejoin that of the as-annealed sample as the
fraction of loaded dislocations is removed.

For the as-rolled samples, the initial values of the dislocation densities
$\rho_{f}$, $\rho_{f}^{*}$ and $\tilde{\rho}_{m}$ were established in the
following way. The initial value of $\rho_{f}$ was computed by evaluating the
above model for the as-annealed material to $\bar{\epsilon}=0.14$ for the
samples rolled at 0 and 90∘ to RD and $\bar{\epsilon}=0.16$ for the sample
rolled at 45∘ to RD. The value of $\rho_{f}$ at the end of these simulated
tensile tests was taken as the starting value for the subsequent predictions
of the rolled tensile samples (see table 3). As no aging was conducted (except
for the short time spent at room temperature between rolling and tensile
testing) the initial value of $\rho_{f}^{*}$ was taken to be zero. This
removes the impact of $\rho_{f}^{*}$ from these predictions. In all cases at
the end of the pre-deformation the model predicts that the material will have
$\rho_{m}=\rho_{m}^{sat}$. We adjust this, however, so that
$\rho_{m}/\rho_{m}^{sat}<1$ prior to re-loading to account for the fact that
on change of strain path many of the dislocations that were active, will no
longer be active (see e.g. [9]). Thus, in this case we have only the initial
value of $\rho_{m}$ as an adjustable parameter. The values of $k_{3}$ and
$k_{4}$ do not impact on the predictions because $\rho_{f}^{*}$ is taken to be
zero.

For the rolled and aged samples we follow a similar process for predicting the
value of $\rho_{f}$ at the pre-strain levels introduced by rolling. The best-
fit values obtained here are reported in table 3. The initial values of
$\rho_{m}$ established for the as rolled samples are retained assuming that
the change in strain path has a larger impact than subsequent aging on the
loss of mobiles dislocations. The initial value of $\rho_{f}$ has been split
into two components, an ‘unloaded’ dislocation density ($\rho_{f}$) and a
‘loaded’ dislocation density ($\rho_{f}^{*}$), where $\rho_{f}^{*}$ is taken
to be much larger than $\rho_{f}$. A good fit could be found by assuming, for
all three test directions, that the initial forest dislocation density in the
as rolled material was partitioned 90% to $\rho_{f}^{*}$ and 10% remaining as
$\rho_{f}$. In the case of Al-Mg samples, it was necessary to account for the
loss of dislocations due to recovery at the aging temperature. Given the
relatively low homologous temperature used for aging here, we envision (as a
starting point) that recovery is small such that loss of dislocation density
can be ignored. For aging at higher temperatures or longer times, this
assumption would need to be revisited.

Parameter | Value | Comment
---|---|---
Material Properties
$b$ | 0.205 nm | Burgers vector
$\mu$ | 80 GPa | Shear modulus
$M$ | 3 | Taylor factor
$\alpha$ | 0.35 | Saada constant
$m$ | 0.02 | Rate sensitivity
Hardening Law Parameters (As-annealed/no strain aging)
$\sigma_{0}$ | 175 MPa | Yield stress (no strain aging)
$k_{1}$ |  $2.0\times 10^{8}$ m-1 | storage term
$k_{2}$ | 11.2 m-1 | annihilation rate
Hardening Law Parameters (Pre-deformed $+$ Aged)
$\eta$ | 5 | generation rate of $\rho_{m}$
$k_{3}$ | $2.4\times 10^{8}m^{-1}$ |
$k_{4}$ | 24 |
$\rho_{m}^{sat}$ |  $1\times 10^{13}$ m-2 |
Table 2: Parameters used in modelling the work hardening response using equations 3, 4 and 6.  Initial $\rho$ | $0^{\circ}$ | $45^{\circ}$ | $90^{\circ}$
---|---|---|---
As rolled
$\rho_{m}$ | $\rho_{m}^{sat}$ | $0.01\rho_{m}^{sat}$ | $0.01\rho_{m}^{sat}$
$\rho_{f}$ |  $1.30\times 10^{14}$, $m^{-2}$ |  $1.35\times 10^{14}$, $m^{-2}$ |  $1.30\times 10^{14}$, $m^{-2}$
$\rho_{f}^{*}$ | 0 | 0 | 0
Rolled and Aged
$\rho_{m}$ | $0.01\rho_{m}^{sat}$ | $0.01\rho_{m}^{sat}$ | $0.015\rho_{m}^{sat}$
Fraction $\rho_{f}$ | 0.1 | 0.1 | 0.1
Fraction $\rho_{f}^{*}$ | 0.9 | 0.9 | 0.9
Table 3: Dislocation densities used in the prediction of the work hardening
response after rolling and after rolling and aging.

The resulting fits to the stress-strain behaviour of the rolled and rolled and
aged samples using the parameters outlined in Tables 2 and 3 is shown in
figure 5. One can see that the model predicts the yield strength and work
hardening behaviour of the samples tested at 0 and 45∘ to RD while predicting
(using the same parameters) the initial yield strength of the sample tested at
90∘ to RD. It can be seen that the initial transient in the work hardening
behaviour following yielding for the sample tested at 45∘ to RD is captured by
the reduced mobile dislocation density.

The model also does a very good job of predicting the yield strength and work
hardening behaviour of the rolled and aged samples, including the initial
transients in the near-yield portion of the stress strain curves from tests at
0 and 45∘ to RD. The work hardening rate at high stress is somewhat over-
predicted for the samples tested at 45 and 90∘ to RD. This likely could be
improved by refining the underlying model but only at the cost of increased
complexity.

Figure 5: a)-c) Stress-strain curves for samples after pre-deformation and
pre-deformation and aging compared to model predictions ($o$). d)-f) shows the
corresponding work hardening rates compared to model predictions. The thick
grey lines in each plot show the behaviour of the as-annealed material.

As a final illustration of the utility of this model, we return to the
‘classic’ case of strain aging under monotonic loading (figure 1), i.e. pre-
deformation in tension followed by aging and re-loading in tension parallel to
the prior loading direction. Figure 6 shows the predictions made using the
above model as a user defined hardening subroutine (VUHARD) within an ABAQUS
finite element simulation. For the as-annealed material, in order to predict
the large upper yield point it was necessary to set the initial mobile
dislocation density to a very small value ($\sim$ 1 m-2). This is in line with
the interpretation [19, 34] that annealing and room temperature aging is
sufficient to lock nearly all available mobile dislocations and sources. The
very high upper yield stress reflects the lack of mobile dislocations in
equation 3. The finite element predicted stress-strain curve under-predicts
the duration of the Lüder’s plateau but this may, in part, be due to the fact
that the strain rate dependency at the Lüder’s front was not accounted for,
instead the macroscopic strain rate was used in equation 3.

Figure 6: Illustration of the use of the constitutive model developed here as
a VUHARD routine within ABAQUS. The model has been applied to the data shown
in figure 1a, i.e. the as-annealed sample and a sample pre-strain in tension,
aged at 150∘C for 10 min.

In the case of prediction for aging a sample at 150∘C for 10 minutes, the
model also does an excellent job of predicting the stress-strain response. In
this case, to obtain the observed results the fraction of forest dislocations
converted to loaded forest dislocations ($\rho_{f}^{*}$) had to be taken to be
lower than that for the rolled and aged samples (70% in this case, compared to
90% in the case of the rolled and aged samples, cf. table 3). In addition, the
mobile dislocation density following aging had to be set to a much lower value
($1\times 10^{-4}\rho_{m}^{sat}$) to give the small upper yield point and
yield point elongation. Again, the impact of the high strain rate at the
Lüder’s front was not considered.

It is interesting to reflect upon the difference in behaviour observed in the
case of the tension-aged-tension and rolled-aged-tension (particularly 0∘ to
RD) tested samples. Clearly the minimal model applied here is unable to
capture all of the complex physics involved in strain path changes and the
results on the stress-strain behaviour, yet it is able to reproduce key
aspects of the stress-stain response. As the level of pre-strain (in rolling)
is increased it may be expected that the model will have difficulty capturing
the increasing load-path-change induced transients on loading [9]. It is also
not obvious why, in tension-aging-tension conditions, the dominant effect
shifts from loaded forest dislocations ($\rho_{f}^{*}$) towards a reduced
mobile dislocation density. Finally, we anticipate that the model presented
here will not account for effects that may arise from backstresses and
kinematic hardening. As mentioned in the introduction, measurements of the
effects of aging when the loading direction is fully reversed (tension-aging-
compression) show a lowering of the yield stress and an extended elasto-
plastic transition [7, 16]. From a macroscopic perspective, a number of models
have been developed which explicitly account for kinematic hardening via
changes to the yield surface (see e.g. [35, 36] for examples of recent work).
Alternatively, microscopic models have been developed (e.g. [37, 28]) which
explicitly build in an evolving back stress at the slip system level, based on
classic concepts of back stress development due to inhomogeneous dislocation
distributions [38]. Incorporating the additional effects induced by strain
aging on such backstress evolution laws would require consideration of the
relaxation of back stresses during strain aging as well as the evolution of a
new set of back stresses on re-loading in the presence of a inhomogeneous
distribution of solute atoms. This is an interesting direction for future
consideration.

##  6 Conclusion

A recently developed model for strain aging [33] has been adapted to predict
the measured stress-strain response of a strain aged ULC steel subjected to a
strain path change, this being more representative of typical application than
the common monotonic pre-deformation in tension, aging, testing in tension
procedure typically used to character strain aging. It is found here that
beyond changes to the yield stress, the work hardening response was changed
significantly upon strain aging followed by a change in strain path. In
particular, it was found that the initial work hardening rate at yield was
significantly higher than that without strain aging. This was attributed,
primarily, to forest dislocations ‘loaded’ by carbon segregation. These
‘loaded’ dislocations are assumed to dynamically recover at a slower rate than
non-loaded dislocations and therefore induce a higher rate of forest
dislocation storage over the initial portion of the stress-strain curve. As
these ‘loaded’ dislocations recover, the work hardening response evolves back
towards that expected in the absence of strain hardening. Testing this
isotropic hardening model under a variety of other strain paths/loading
conditions would be an ideal way to test the limits of the model and to
identify conditions (like fully reversed strain path changes) where it is
expected that the model will need to be improved.

##  7 Acknowledgements

The authors would like to acknowledge funding from the Canadian Natural
Science and Engineering Research Council of Canada for funding and the
provision of materials by ArcelorMittal Dofasco.

## References

  * [1] V. Ballarin, A. Perlade, X. Lemoine, O. Bouaziz, S. Forest, Mechanisms and modeling of bake-hardening steels: Part II. Complex loading paths, Metallurgical and Materials Transactions A: Physical Metallurgy and Materials Science 40 (6) (2009) 1375–1382.  doi:10.1007/s11661-009-9812-6.
  * [2] E. Pereloma, I. Timokhina, Bake hardening of automotive steels, 2017.  doi:10.1016/B978-0-08-100638-2.00009-2.
  * [3] A.K.De., S. Vandeputte, B. Soenen, B. De Cooman, Effect of grain size on the static strain aging of ULC-bake hardening steel, Zeitschrift fur Metallkunde 95 8 (2004) 713–717.
  * [4] A. K. De, K. De Blauwe, S. Vandeputte, B. C. De Cooman, Effect of dislocation density on the low temperature aging behavior of an ultra low carbon bake hardening steel, Journal of Alloys and Compounds 310 (1-2) (2000) 405–410.  doi:10.1016/S0925-8388(00)00973-7.
  * [5] S. Berbenni, V. Favier, X. Lemoine, M. Berveiller, A micromechanical approach to model the bake hardening effect for low carbon steels, Scripta Materialia 51 (4) (2004) 303–308.  doi:10.1016/j.scriptamat.2004.04.031.
  * [6] V. Ballarin, M. Soler, A. Perlade, X. Lemoine, S. Forest, Mechanisms and modeling of bake-hardening steels: Part I. Uniaxial tension, Metallurgical and Materials Transactions A: Physical Metallurgy and Materials Science 40 (6) (2009) 1367–1374.  doi:10.1007/s11661-009-9813-5.
  * [7] G. Saada, T. Kruml, Removal of plastic instabilities by reversal of the applied stress, Philosophical Magazine 93 (2013).  doi:10.1080/14786435.2012.702232.
  * [8] G. Vincze, E. F. Rauch, J. J. Gracio, F. Barlat, A. B. Lopes, A comparison of the mechanical behaviour of an aa1050 and a low carbon steel deformed upon strain reversal, Acta Materialia 53 (2005).  doi:10.1016/j.actamat.2004.10.046.
  * [9] D. V. Wilson, P. S. Bate, Influences of cell walls and grain boundaries on transient responses of an if steel to changes in strain path, Acta Metallurgica Et Materialia 42 (1994).  doi:10.1016/0956-7151(94)90127-9.
  * [10] J. H. Schmitt, J. V. Fernandes, J. J. Gracio, M. F. Vieira, Plastic behaviour of copper sheets during sequential tension tests, Materials Science and Engineering A 147 (1991).  doi:10.1016/0921-5093(91)90840-J.
  * [11] I. J. Beyerlein, C. N. Tomé, Modeling transients in the mechanical response of copper due to strain path changes, International Journal of Plasticity 23 (2007).  doi:10.1016/j.ijplas.2006.08.001.
  * [12] P. S. Bate, I. Brough, S. Morse, Static recovery and the orthogonal strain path change effect in if steel, Materials Science Forum 550 (2007).  doi:10.4028/www.scientific.net/msf.550.141.
  * [13] R. Sowerby, Y. Tomita, On the bauschinger effect and its influence on U.O.E. pipe making process, International Journal of Mechanical Sciences 19 (6) (1977) 351–359.  doi:10.1016/0020-7403(77)90088-1.
  * [14] R. A. Elliot, E. Orowan, T. Udoguchi, A. S. Argon, Absence of yield points in iron on strain reversal after aging, and the Bauschinger overshoot, Mechanics of Materials 36 (11) (2004) 1143–1153.  doi:10.1016/j.mechmat.2004.04.001.
  * [15] D.N.Williams, Interaction between the Bauschinger effect and strain aging, Metall. Trans., A; 11 (September) (1980) 1629–1631.
  * [16] Q. Lai, C. W. Sinclair, W. J. Poole, The anisotropic kinetics of strain aging in a martensitic steel, Canadian Metallurgical Quarterly 57 (1) (2018) 20–27.  doi:10.1080/00084433.2017.1384211.
  * [17] G. Jun, W. F. Hosford, Flow behavior of an aluminum-killed steel after tensile prestraining and strain-aging, Metallurgical Transactions A 17 (1986).  doi:10.1007/BF02650093.
  * [18] S. Hiwatashi, S. Yonemura, Anisotropic Increase in Yield Strength by Straining and Baking of Bake-Hardenable Steel, Materials Science Forum 495-497 (2009) 1485–1492.  doi:10.4028/www.scientific.net/msf.495-497.1485.
  * [19] G. T. Hahn, A model for yielding with special reference to the yield-point phenomena of iron and related bcc metals, Acta Metallurgica 10 (8) (1962) 727–738.  doi:10.1016/0001-6160(62)90041-X.
  * [20] M. Smith, ABAQUS/Standard User’s Manual, Version 6.14, Dassault Systèmes Simulia Corp, United States, 2014.
  * [21] N.Louat, On the theory of the Portevin-Le Chatelier effect, Scripta Metallurgica 15 (c) (1981) 1167–1170.
  * [22] A. Cottrell, Theory of dislocations, Progress in Metal Physics 4 (1953) 205–264.  doi:10.1016/0502-8205(53)90018-5.
  * [23] A. K. De, B. C. De Cooman, S. Vandeputte, Kinetics of strain aging in bake hardening ultra low carbon steel-a comparison with low carbon steel, Journal of Materials Engineering and Performance 10 (5) (2001) 567–575.  doi:10.1361/105994901770344719.
  * [24] C. S. Lee, B. J. Duggan, Simple theory for the development of inhomogeneous rolling textures, Metallurgical Transactions. A 22 A (1991).  doi:10.1007/BF02851357.
  * [25] W. F. Hosford, R. M. Caddell, Metal forming: Mechanics and metallurgy, 2011.  doi:10.1017/CBO9780511976940.
  * [26] U. F. Kocks, H. Mecking, Physics and phenomenology of strain hardening: The fcc case, Progress in Materials Science 48 (2003).  doi:10.1016/S0079-6425(02)00003-8.
  * [27] M. van Riel, A. H. van den Boogaard, Stress-strain responses for continuous orthogonal strain path changes with increasing sharpness, Scripta Materialia 57 (5) (2007) 381–384.  doi:10.1016/j.scriptamat.2007.05.005.
  * [28] W. Wen, M. Borodachenkova, C. N. Tomé, G. Vincze, E. F. Rauch, F. Barlat, J. J. Grácio, Mechanical behavior of low carbon steel subjected to strain path changes: Experiments and modeling, Acta Materialia 111 (2016).  doi:10.1016/j.actamat.2016.03.075.
  * [29] E. V. Nesterova, B. Bacroix, C. Teodosiu, Microstructure and texture evolution under strain-path changes in low-carbon interstitial-free steel, Metallurgical and Materials Transactions A: Physical Metallurgy and Materials Science 32 (10) (2001) 2527–2538.  doi:10.1007/s11661-001-0042-9.
  * [30] H. Mecking, U. F. Kocks, Kinetics of flow and strain-hardening, Acta Metallurgica 29 (11) (1981) 1865–1875.  doi:10.1016/0001-6160(81)90112-7.
  * [31] E. Jones, T. Oliphant, P. Peterson, et al., SciPy: Open source scientific tools for Python (2001–).
URL http://www.scipy.org/

  * [32] L. Kubin, Y. Estrin, Evolution of dislocation densities and portevin-Le chatelier effect, Acta. Metall. Mater. 38 (5) (1990) 697–708.
  * [33] S. Medrano, C. W. Sinclair, Transient strain age hardening of Al – Mg alloys, Materialia 12 (March) (2020).  doi:10.1016/j.mtla.2020.100796.
  * [34] J. J. Gilman, W. G. Johnston, Dislocation velocities, dislocation densities, and plastic flow in lithium fluoride crystals, Journal of Applied Physics 30 (2) (1959) 129–144.  doi:10.1063/1.1735121.
  * [35] T. Mánik, B. Holmedal, O. S. Hopperstad, Strain-path change induced transients in flow stress, work hardening and r-values in aluminum, International Journal of Plasticity 69 (2015).  doi:10.1016/j.ijplas.2015.01.004.
  * [36] J. Qin, B. Holmedal, K. Zhang, O. S. Hopperstad, Modeling strain-path changes in aluminum and steel, International Journal of Solids and Structures 117 (2017).  doi:10.1016/j.ijsolstr.2017.03.032.
  * [37] K. Kitayama, C. N. Tomé, E. F. Rauch, J. J. Gracio, F. Barlat, A crystallographic dislocation model for describing hardening of polycrystals during strain path changes. application to low carbon steels, International Journal of Plasticity 46 (2013).  doi:10.1016/j.ijplas.2012.09.004.
  * [38] H. Mughrabi, Dislocation wall and cell structures and long-range internal stresses in deformed metal crystals, Acta Metallurgica 31 (1983).  doi:10.1016/0001-6160(83)90007-X.