Mg2012: 9th International Conference on Magnesium Alloys and their Applications
W.J. Poole and K.U. Kainer
A NEW MODEL TO PREDICT GRAIN NUCLEATION DURING
DYNAMIC RECRYSTALLIZATION
Abhijit Brahme1, Yauheni Staraselski1, Raja K. Mishra2 and Kaan Inal1
1
University of Waterloo; 200 University Avenue West, Waterloo, Ontario, N2L 3G1, Canada
2
General Motors Research & Development Centre; Warren, MI 48090 USA
Keywords: Dynamic recrystallization, Texture, Nucleation, CPFEM
Abstract
Deformation in metals and alloys is accompanied by high hardening rates and high dislocation
content. Static annealing of deformed metals or forming operations at elevated temperatures can
lead to static and dynamic recrystallization (SRX and DRX). The resultant texture and hence the
properties of such a material are determined by the nucleation and growth of recrystallized grains
conditioned by the deformation. Accurate prediction of nucleation in the deformed state is essential
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nucleation of recrystallized grains based on local gradient in the dislocation density tensor in the
deformed material. The Nye tensor, which measures local lattice curvature and is an indicator
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deformation mechanisms as well as texture and its evolution. Nucleation is postulated to occur in
regions of the microstructure that have high (locally) stored energy adjacent to a low Nye tensor
(relatively undeformed) zone. Texture resulting from different starting microstructure are simulated
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Introduction
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phenomena during deformation of metals. When these alloys are deformed at elevated temperatures,
dynamic recrystallization (DRX) [1] can be observed. DRX accompanying deformation can
enhance formability beyond what can be accomplished due to increased slip activity and grain
boundary sliding at elevated temperatures. DRX removes new defects continuously and extends
the stress-strain curve to high strain values without hardening. Once the material is loaded at
elevated temperatures deformation is accommodated either by slip or twinning till a critical value
is reached. After this critical value, the material starts showing some degree of recrystallization,
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of a new undeformed grain in a highly deformed zone (nucleation) and (b) growth of the nucleus
at the expense of deformed matrix.
Unlike static recrystallization (SRX), the dislocation content during DRX evolves with time both
in the matrix and the recrystallized region. Thus understanding and characterization of nucleation
and growth by modeling the dislocation content by crystal plasticity methods provides a method
to model DRX. DRX has been modeled in the past using cellular automata (CA) [2,3] methods
that use a phenomenological model such as Kocks and Mecking law [3] to calculate the change
in dislocation density which cannot account for local evolution of deformed microstructure that
determine the nucleation sites. This study presents a coupled FE model to compute local evolution
of deformation in a sample of AA5754 aluminum sheet that uses microstructural data from EBSD
maps as input and accounts for the known operative deformation mechanisms. A nucleation
criterion is developed and implemented into an existing in-house FE model to predict the new
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in the microstructure reconstruction and Nye tensor determination for basal and nonbasal slip
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systems in HCP Mg alloys are discussed to make this approach applicable to Mg alloys that need
to be processed at elevated temperatures for enhanced formability where DRX becomes important.
Nucleation Criterion
Nucleation:
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of a criterion to designate a subgrain as a nucleus is very important in this approach. Various
criteria have been proposed in literature for initiation of DRX. Most of the nucleation schemes
are statistical, relying on critical value of wT /wV >@ZKLFKGHQRWHVDQLQÀHFWLRQSRLQWLQWKHRQ
hardening rate (T)-V curve, strain rate [5], etc. with the nuclei being placed randomly along grain
boundaries [6]. Cram et al. [7] proposed a criterion based on critical subgrain size dependent on
the total strain in the material, contrary to DRX observations. In this work, a criterion based on a
critical difference (jump) in dislocation content for a potential nucleus to be viable is presented to
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higher driving force for boundary migration. The driving force can be provided by difference in
stored energy where the stored energy depends on the dislocation content at any strain and strain
rate. The zone with low dislocation content becomes a nucleus when it is surrounded by region of
high dislocation content with a steep gradient (or boundary).
Evaluation of dislocation content:
The dislocation content at any given point in the microstructure is calculated by employing the
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1
b
D = ’ u FP
(1)
where )3 is the plastic deformation gradient, which can be calculated from the accumulated shear
during deformation and Eis burgers vector. Crystal plasticity based FE models have been developed
to accurately characterize the evolution of deformation in various alloy systems like HCP [9] and
FCC [10] metals where different deformation mechanisms are operational. Details of such FE
and crystal plasticity constitutive model formulations can be found in [9,10] and are not repeated
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work (see [10] for details of the code). The FE code provides the deformation gradient, F, at each
integration point (IP) in the FE mesh which can be constructed from real microstructures using
electron backscatter diffraction data. The plastic part of the deformation gradient (FP) is calculated
from the total deformation gradient (F). The DDT is calculated at each IP in the mesh by taking the
curl of FP using Eq. 1. The DDT can then be used to calculate the average jump (d||D||) in the DDT
for the neighborhood of each IP.
1 n
d D = ¦ Di D j
n j =1
(
)
(2)
where i is the current IP, j is neighbor IP and n is the number of nearest neighbors. d||D|| is evaluated
for each IP in the microstructure after each time step. If d||D|| in a given IP is higher than a threshold
value, then the element is tagged as a nucleus, with the lower D value region representing the
recrystallized grain and gradient corresponding to the mobile boundary that would consume the
high D zone as recrystallization proceeds. This procedure is repeated by updating D and d||D|| after
each time step in the FE code.
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Results and Discussion
Figure 1 shows the evaluated DDT on a synthetic microstructure reconstructed using EBSD data
from an annealed AA5754 aluminum sheet using procedure described in Brahme et al. [11]. Since
the actual reconstructed microstructure by this method is not unique at the nearest neighbor grain
level, two different microstructures that have the same statistical grain size, orientation distribution
and misorientation distribution were constructed as shown in Figs. 2(a) and (b), marked M-I and
M-II respectively. The X and Y-axis show the sample X and Y direction and the Z-axis shows the
evaluated ||D|| (||D__ 6_D_), in arbitrary units, for each element in the microstructure. The value of
||D|| depends on the local value of the deformation gradient ()3); higher values of ||D|| correspond to
higher dislocation content but do not imply a higher likelihood of nucleation. The local mismatch
or jump in the dislocation density is calculated and is reported as an average value d||D|| at each
element and this is used as an indicator of nucleation probability.
The two microstructures were subjected to uniaxial tension along the RD (X-axis). The material
can be subjected to SRX by removing the strain and then subjecting to annealing or be subjected to
DRX by holding the sample at elevated temperature if the FE simulation is done for the appropriate
deformation temperature to calculate DThe methodology of nucleation selection at a point during
deformation after 20 % elongation (~18% true strain) along RD is chosen for the analysis below.
The dislocation content (||D||) and the average jump in dislocation content (d||D||) are evaluated
at each integration point in the material. The resultant value of d||D||, in relative units, is shown
in Figures 3DDQGEIRU0,DQG0,,UHVSHFWLYHO\&RPSDULQJ¿JXUHVDDQGELWFDQEH
seen that the two microstructures lead to a very different distributions of d||D||. This is due to the
differences in the local microstructure which results in different slip system activity and hence
in different local deformation gradient. A high value of d||D|| in a particular location implies that
such a region in the microstructure has higher mismatch in D tensor. That is, the region has higher
D and is surrounded by regions with lower D or vice versa. The second case corresponds to a
possible nucleation site for a recrystallized grain as it implies that the region of interest has lower
dislocation content than its surrounding matrix and can grow to reduce stored work, thus lowering
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for a new grain. By selecting an appropriate value of d||D|| one can have a method of identifying
nucleation sites. If a region in the microstructure has d||D|| higher than a certain predetermined
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Effect of d||D|| threshold
Figure 4 shows the texture plots corresponding to the orientations of the nucleated grains of the
recrystallized microstructure, selected using different threshold values for d||D|| (in the range
d||D||=5E-05 to d||D||=4E-04 in relative units) on M-I. Figure 4 (f) has the highest threshold and
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a few poles. Figure 4 (a) has smallest threshold and hence has the most spread in orientations of
the nuclei. Clearly values of d||D||=3E-04 and 4E-04 provide very few nuclei and the d||D||=5E-05
provides too many – the lower and upper bounds. The actual value of the threshold should be in
between these bounds. For purposes of comparison of the recrystallization texture between the two
microstructures (M-I and M-II) a value of d||D||threshold=1.5E-04 is chosen.
Texture Comparison between two microstructures
For comparison of the prediction of the orientations of the nucleated grains produced from the two
microstructures, M-I and M-II, d||D|| was calculated and a threshold value for d||D||=1.5E-04 was
used to identify nuclei. The resultant nuclei texture is shown in Figure 5. Comparison with the
texture for the same value of threshold for microstructure M-I (Fig. 4c) shows that the predicted
texture for both microstructures are similar despite both of them having very different distributions
of d||D|| as can be seen in Fig. 3.
71
Extension of the model to Mg alloys
Mg alloy sheets such as AZ31 have a hexagonal crystal structure and the microstructure
reconstruction from EBSD data for HCP symmetry has already been developed. These sheets
exhibit strong basal texture and deformation occurs by twinning as well as basal and nonbasal slip.
twins and the distribution of different slip systems in the grain interior and grain boundaries [12].
Khosravani et al. [13] have developed a method to measure the Nye tensor for basal and nonbasal
dislocations during deformation of AZ31 sheets which can provide important data for calibration
of the threshold d||D|| for SRX and DRX. Thus the method discussed above for FCC aluminum
alloy can be extended to simulate SRX and DRX in Mg alloys.
Summary
The Nye Tensor, ||D||, as a measure of local dislocation density variations, can be used to compute
recrystallization by coupling the crystal plasticity simulation with a nucleation condition
determined by a threshold value of local mismatch (d||D||) in dislocation density. The distribution
of d||D|| depends on the microstructure of the material. Different microstructures having different
distribution of d||D|| lead to similar nuclei texture for the same threshold value of local mismatch
in static recrystallization simulation. The method is suitable to study the dynamic evolution of
deformation structure and DRX. By calibrating the model against experimentally measured
dislocation density values. The proposed approach is demonstrated for AA5754 and can be easily
extended to Mg alloys.
References
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[12]
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T. Sakai and J.J. Jonas, Acta Metall. 32 p. 189 (1984).
R.L. Goetz and V. Seetharaman Scripta Mat. 38 p 405 (1998).
G. Kugler and R. Turk, Acta Mat. 52 p 4659 (2004).
E.I. Poliak and J.J. Jonas Acta Mat. 44 p. 127 (1996).
P. Peczak and M.J. Luton Phil. Mag. B 70 p 817 (1994).
R Ding and Z.X. Guo Acta Mat. 49 p 3163 (2001).
D.G. Cram et al. Acta Mat. 57 p 5218 (2009).
J.F. Nye, Acta Mettal. 1 p. 153 (1953).
A. Izadbakhsh et al., Comp. Mat. Sci. 50 p 2185 (2011).
J. Rossiter et al., Int. J. Plas. 26 p 1702 (2010).
A. Brahme et al., Scripta Mat. 55 p 75 (2006).
É. Martin et al., Phil. Mag. 91 p. 3613 (2011).
A. Khosravani et al., TMS Annual Meeting, Orlando, 2012
72
Table of Figures
Figure 1: Showing ||D|| along a plane in the microstructure. The possible nucleation sites are not
necessarily the ones with high ||D|| but the ones with biggest jump locally
Figure 2: Two different microstructures generated (a) I and (b) II. Both have similar overall
ODF and MDF but different local arrangement of orientation. The shades are indicative of their
crystallographic orientation.
)LJXUH&RQWRXUVVKRZLQJWKHORFDOPLVPDWFKLQ__Į__G__Į__IRUGLIferent microstructures (a)
73
M-I and (b) M-II respectively.
Figure 4: Predicted nuclei texture for Microstructure II with different threshold (d|| D ||) viz. (a)
d||D || = 5E-05, (b) d||D || = 1E-04, (c) d||D || = 1.5E-04, (d) d||D || = 2E-04, (e) d||D || = 3E-04
and (f) d||D || = 4E-04 respectively.
)LJXUH3UHGLFWHGQXFOHLWH[WXUHIRU0LFURVWUXFWXUH,ZLWKG__Į__ (7KHORFDOWH[WXUH
YDULDWLRQDQGWKHORFDOPLVPDWFKRI__Į__RIIRUERWKPLFURVWUXFWXUHDUHGLIIHUHQW7KHSUHGLFWHG
texture for the two microstructures with similar thresholds is similar (compare to Fig. 4c).
74