Plastic anisotropy of electro-deposited pure α-iron with sharp crystallographic <1 1 1>// texture in normal direction: Analysis by an explicitly dislocation-based crystal plasticity model

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Plastic anisotropy of electro-deposited pure a-iron with sharp
crystallographic <111>// texture in normal direction: Analysis
by an explicitly dislocation-based crystal plasticity model
Alankar Alankara,b,c,⇑, David P. Fieldb, Dierk Raabec
aMST-8, Materials Science and Technology Division, MS G755, Los Alamos National Laboratory, Los Alamos, NM 87545, USA
bSchool of Mechanical and Materials Engineering, Washington State University, Box 642920, Pullman, WA 99164, USA
cMicrostructure Physics and Alloy Design, Max-Planck Institut für Eisenforschung GmbH, 40235 Düsseldorf, Germany
a r t i c l ei n f o
Article history:
Received 12 September 2012
Received in final revised form 5 February
2013
Available online 5 April 2013
Keywords:
Crystallographic texture
a-Iron
Dislocations
Kink-pairs
Metal sheet forming
a b s t r a c t
We present a single crystal plasticity model based on edge and screw dislocation densities
for body centered cubic (bcc) crystals. In a bcc crystal screw dislocations experience high
lattice friction due to their non-planar core. Hence, they have much slower velocity com-
pared to edge dislocations. This phenomenon is modeled by accounting for the motion of
screw dislocations via nucleation and expansion of kink-pairs. The model, embedded as a
constitutive law into a crystal plasticity framework, is able to predict the crystallographic
texture of a bcc polycrystal subjected to 70%, 80% and 90% thickness reduction. We perform
a parametric study based on the velocities of edge and screw dislocations to analyze the
effect on plastic anisotropy of electro-deposited pure iron with long needle-shaped grains
having sharp crystallographic <111>//ND texture (ND: normal direction). The model
shows a large change in the r-value (Lankford value, planar anisotropy ratio) for pure iron
when the texture changes from random to <111>//ND. For different simulated cases where
the crystallites have an orientation deviation of 1?, 3? and 5?, respectively, from the ideal
<111>//ND axis, the simulations predict r-values between 4.0 and 7.0 which is in excellent
agreement with data observed in experiments by Yoshinaga et al. (ISIJ Intern., 48 (2008)
667–670). For these specific orientations of grains, we also model the effect of long needle
shaped grains via a procedure that excludes dislocation annihilation.
? 2013 Elsevier Ltd. All rights reserved.
1. Introduction
During the last two decades efforts were made to explicitly incorporate dislocation density-based constitutive laws into
crystal plasticity finite element models e.g. Alankar et al. (2012a, 2009), Aoyagi et al. (2013), Arsenlis and Parks (2002),
Arsenlis and Tang (2003), Cheong and Busso (2004), Evers et al. (2004), Erieau and Rey (2004), Lim et al. (2011), Ma et al.
(2006), Mayeur et al. (2011), Ortiz et al. (2000) and Shanthraj and Zikry (2011). A detailed review of dislocation density based
crystal plasticity models has been done by Roters et al. (2010). While all the aforesaid approaches are phenomenological,
some of these use dislocation density as power law dependent variable (e.g. Lim et al. (2011)), the others have them as self
sustained geometric loops (e.g. Arsenlis and Parks (2002)). Dislocation density based approach is employed not only for
understanding single crystal behavior but also for simulations of polycrystal plasticity. See e.g. Austin and McDowell
(2011), Erieau and Rey (2004) and Lim et al. (2011). However, when studying macroscopic plastic anisotropy in metal form-
ing processes, most finite element (FE) approaches use viscoplastic crystal plasticity or even Taylor-based approximations
0749-6419/$ - see front matter ? 2013 Elsevier Ltd. All rights reserved.
http://dx.doi.org/10.1016/j.ijplas.2013.03.006
⇑Corresponding author at: MST-8, Materials Science and Technology Division, MS G755, Los Alamos National Laboratory, Los Alamos, NM 87545, USA.
Tel.: +1 505695 5542; fax: +1 505667 8021.
E-mail address: alankar@lanl.gov (A. Alankar).
International Journal of Plasticity 52 (2014) 18–32
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rather than dislocation based constitutive laws. Examples are metal forming simulations based on Taylor-type fixed con-
straints (FC) type or relaxed constraints (RC) type (Raphanel and Van Houtte, 1985; Van Houtte, 1988; Yoshinaga et al.,
2008), or self consistent schemes in conjunction with FEM (Segurado et al., 2012) or viscoplastic crystal plasticity FE models
(Beaudoin et al., 1994; Beaudoin et al., 1993; Kalidindi and Schoenfeld, 2000; Neale, 1993; Raabe and Roters, 2004; Raabe
et al., 2005; Tikhovskiy et al., 2008; Xie and Nakamachi, 2002; Zhao et al., 2001). Phenomenological approaches e.g. by
Yoshida et al. (2007) have been used for showing the effect of crystallographic texture on formability. Constitutive relations
used in a number of aforementioned crystal plasticity model use power law based relations between resolved shear stress
(RSS) and shear strain rates on slip systems. Such power law based constitutive relations can be fitted extremely well against
the experimentally observed stress–strain response of metals in a wide range of temperature and strain rate. However, the
effect of dislocation density evolution on the formability of metals is not commonly reported. Moreover, for incorporating
fundamentals of microstructure evolution e.g. strength of dislocation reactions based on crystal structures (Alankar et al.,
2012b; Devincre et al., 2008; Madec and Kubin, 2003; Queyreau et al., 2009), crystal plasticity models explicitly based on
dislocation densities are required (Erieau and Rey, 2004). Here, we demonstrate that by using a crystal plasticity model in
conjunction with a novel dislocation-based constitutive model for the prediction of crystallographic texture-related anisot-
ropy, macroscopic forming operations can be understood and designed on a sound microstructural basis. This involves
specifically incorporating crystalline plastic anisotropy not only at the kinematic level (texture) but also at the constitutive
dislocation behavior level. We show that such dislocation-based crystal plasticity simulation concepts have advantages over
conventional viscoplastic methods that require empirical parameters as well as biased or even speculative considerations on
internal constraints and micro–macro homogenization.
Most crystal plasticity studies have so far been performed for fcc metals (e.g. Han et al., 2007; Field and Alankar, 2011)
and only a few approaches were made to predict specific bcc crystal mechanics (Erieau and Rey, 2004; Lee et al., 1999; Ma
et al., 2007; Stainier et al., 2002; Stainier et al., 2003). Some of the published bcc models make no principal difference be-
tween the constitutive framework used for fcc crystals and for non-fcc crystals except for the different Schmid factors of
the underlying slip systems (Kitayama et al. 2012; Raphanel and Van Houtte, 1985).
In a bcc crystal, {112} and {123} slip planes have been reported to be active with <111> slip directions (Barrett et al.,
1937; Gough, 1928). Some studies suggest that only {110} and {112} planes play an active role at lower temperatures
and {123} slip occurs only at higher temperatures (Raabe, 1995; Sesták and Seeger, 1978a, 1978b). Other sources indicate
that at room temperature only {110} glide planes are active based on the argument that the slip observed on {112} and
{123} planes is actually a composite slip on two alternating {110} planes e.g. Chen and Maddin (1954). Therefore, in this
work we use {110} <111> and {112} <111> slip systems.
Besides these kinematic aspects another key difference between the crystal plasticity of fcc crystals and bcc crystals is
that in bcc materials non-Schmid behavior occurs (Taylor, 1928). In the present work we, therefore, focus on bcc metals
which show a non-uniqueness of the shear planes associated with a/2 <111> screw dislocations due to their non-planar core
structure which extends into 3 {110} and 3 {112} planes that pertain to the same <111> zone (Ito and Vitek, 2001; Vitek,
1974, 1976). The low energy non-planar core structure causes deep Peierls valleys along the glide direction. For minimizing
the energy corresponding to the line length, the screw dislocations hence tend to reside in the form of long segments inside
the Peierls valleys (Vitek, 1974, 1976). For the further movement of the screw dislocations, it is necessary that the non-planar
cores are constricted in a way that the energy of constriction is in accordance with the Peierls stress required for the motion
of screw dislocation. The spread-out core of screw dislocations in bcc metals is responsible for the non-Schmid behavior
(Vitek, 1974). Edge dislocations, on the other hand, do not face such a high Peierls stress. Therefore, long screw dislocation
lines and short edge dislocation lines are characteristic of bcc metals deformed at moderate and low temperatures (Akhtar
and Teghtsoonian, 1975; Cai, 2001). The actual motion of such long screw segments in the presence of high Peierls barriers
occurs via the formation and expansion of double kink configurations (Hirth and Lothe, 1992; Seeger and Schiller, 1962). This
behavior is common to all bcc metals irrespective of the details of the inter-atomic bonding, owing to the fact that a/2 <111>
screw dislocations are formed along a threefold symmetry axis. Similar behavior has been identified in HCP metals in which
screw dislocations on prismatic planes have a non-planar core structure spread out on basal and first order pyramidal planes.
The motion of screw dislocations on prismatic and pyramidal planes in HCP crystals can, therefore, also be described via the
formation of double kink screw dislocations (Alankar et al., 2011; Monnet et al., 2004; Srivastava et al., 2013).
In this work we present a micromechanical model for bcc crystals considering the individual behavior of screw and edge
dislocations, as outlined above. The effect of different edge and screw dislocation kinetics on metal deformation was also
pointed out by Stainier et al. (2002). Recently, Acharya and Chapman (2012) have pointed out that there is a strong effect
of dislocation structures on the macroscopic plastic anisotropy. We employ the edge and screw dislocation density based
constitutive model for analyzing the physical origin of extremely high r-values that were observed in electrodeposited
<111>//ND textured pure a-Fe sheets by Yoshinaga et al. (2008). The r-value which is also referred to as Lankford value de-
scribes the ratio between the in-plane deformation component and the deformation component in through-thickness direc-
tion (r-value = ew=et). A high r-value indicates that material flow occurs primarily parallel to the plane surface in metal
forming operations.
A study of r-values of electrodeposited <111>//ND was performed by (Yoshinaga et al., 2008) using a Taylor-based
homogenization method (Van Houtte, 1988) for determining the crystalline anisotropy. However, the crystallographic model
was not suited to predict the extremely high anisotropy values observed in the experiments (Yoshinaga et al., 2008). We
study the effect of different velocities of edge and screw dislocations on the plastic anisotropy of electrodeposited
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<111>//ND textured pure a-Fe sheets. It enables us to understand and predict the physical processes underlying crystalline
anisotropy both from an elementary dislocation level (single slip system behavior) and crystal level (multi slip system
behavior; crystallographic texture) perspective. The interaction of these factors (individual dislocation kinetics including dif-
ferences between screw and edge components; intra-grain multislip conditions; inter-grain constraints) cannot be under-
stood via polycrystal experiments alone. Specifically, by applying the present model to the phenomenon of giant plastic
anisotropy, we can demonstrate that the proper incorporation of elementary dislocation kinetic features can have substantial
influence on the correct prediction of macroscopic forming characteristics such as deformation-induced plastic anisotropy.
In the following sections we present the constitutive framework, simulation results, discussions and conclusions.
2. Constitutive framework
The kinematics of the shear rate on each slip system can be written in terms of a generalized Orowan form. The total plas-
tic shear strain rate is given by (Arsenlis and Parks 2002; Arsenlis and Tang, 2003; Alankar et al. 2009, 2012a; Alankar and
Field, 2012):
_ ca¼ baðqa
e? va
eþqa
s? va
sÞsignðsaÞð1Þ
where bais the magnitude of the Burgers vector and sais the resolved shear stress (RSS) on slip system a. qa
dislocation density,qa
age velocity for the screw type dislocations.
Fig. 1 shows the asymmetric arrangement of the slip planes associated with the slip direction <111>. On {112} planes the
slip resistance along the twin direction is lower than that for slip along anti-twin directions due to the non-Schmid effect in a
bcc crystal (Vitek, 1974, 1976). The twin and anti-twin directions are denoted by ‘T’ and ‘A’ respectively. The slip systems
along with the twin and anti-twin directions are shown in Table 1.
eis the edge
sis the aver-
sis the screw dislocation density,? ma
eis the average velocity for edge type dislocations and? ma
2.1. Dislocation multiplication
The rates for the multiplication and annihilation of edge and screw dislocations are adopted from the work of Arsenlis and
Parks (2002) and from the work of Alankar et al. (2009, 2011, 2012a). We describe the edge and screw dislocation densities
through the expansion of dislocation loops that proceed from Frank-Read sources. This means that the edge dislocation den-
sity increases due to moving screw dislocations and the screw dislocation density increases due to moving edge dislocations
as given by the following equations:
_ qa
e;mul¼ 2qa
s? va
?la
s
s
ð2aÞ
_ qa
s;mul¼ 2qa
e? va
?la
e
e
ð2bÞ
where?la
eand?la
srepresent average segment lengths for the edge and screw type dislocations, respectively.
Fig. 1. Orientation of the {110} and {112} planes that pertain to the same [111] crystallographic zone. The sense of ‘twin’ and ‘anti-twin’ are represented
by the symbols ‘T’ and ‘A’ respectively. (After Lee et al. (1999)).
20
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2.2. Dislocation annihilation
Two dislocations of opposite sign annihilate each other when they come within a critical distance. The annihilation rates
are given by (Arsenlis and Parks 2002; Arsenlis and Tang, 2003; Alankar et al. 2009, 2011, 2012a):
_ qa
e;ann¼ ?ðqa
eÞ2Ra
e? va
e
ð3aÞ
_ qa
s;ann¼ ?ðqa
eand Ra
sÞ2Ra
s? va
s
ð3bÞ
where Ra
tively. This means that the critical radius is the minimum distance up to which two opposite signed dislocations can ap-
proach while gliding on the same plane before they spontaneously annihilate.
For both, edge and screw dislocations, Arrhenius type equations for the velocity are used. The edge dislocations are as-
sumed not to be affected by the Peierls stress. The edge dislocation velocity is given by the equation:

sare the critical radii for edge and screw dislocations, below which spontaneous annihilation occurs, respec-
? va
e¼ ? ve;0exp ?F0
kBT
1 ?
jsaj
e;0þ Sa
sa
!pe
!qe
0
@
1
A
ð4aÞ
where ? ve;0is a reference velocity for edge dislocations, F0the free energy of the activation of motion of edge dislocations, kB
the Boltzmann constant, se,0the lattice friction stress acting on edge dislocations and T the absolute temperature. pe and qe
are material coefficients such that 0 6 pe 6 1.0 and 1.0 6 qe 6 2.0. Sais the slip resistance and sais the resolved shear stress
on slip systema. Due to high Peierls stress acting on screw dislocations, we describe screw dislocation motion in terms of the
formation of kink pairs. The velocity of a screw dislocation segment (? va
kink pairs. It is given by Dorn and Rajnak (1964), Duesbery (1969) Guyot and Dorn (1967) Hirth and Hoagland (1993) Tang
et al. (1998) Xu and Moriarty (1998):
?
where F0,kinkis the free energy of the formation of a kink-pair, ps and qs are material coefficients such that 0 6 ps 6 1 and
1 6 qs 6 2. lais the length of the straight screw dislocation segment. l0is the critical length for kink pair nucleation. mDis
the Debye frequency, spis the Peierls stress, and bais the magnitude of the Burgers vector. Stainier et al. (2003) proposed
temperature and strain rate dependent constitutive form of Peierls stress. Since, in the present work we performed all the
simulations at room temperature, we use a constant value of the Peierls stress. The term mDba/l0describes the attempt fre-
quency for kink pair formation and the term la=l0describes the number of competing sites for kink pair formation on the
segment la(Kubin et al., 1998; Tang et al., 1998).
s) is governed by thermally activated nucleation of
? va
s¼ bala
l0
tDba
l0
exp
?F0;kink
kBT
1 ?
jsaj
sP
?ps
??qs
!
ð4bÞ
2.3. Forest hardening
The slip resistance Saon a slip system a is given by a modified Taylor type equation (Franciosi et al., 1980) as shown be-
low. The forest hardening is applicable only for edge dislocations in bcc metals since screw dislocations have very low veloc-
ity as compared to edge dislocations and do not form a junction with forest dislocations (Monnet et al., 2004). The slip
resistance Saon a slip system a is given by:
Table 1
Slip directions (ma
0) and slip plane normals (na
0) in bcc crystals.
Slip system (a)(ma
0)(na
0)Slip system (a)Label(ma
0)(na
0)
1
2
3
4
5
6
7
8
9
10
11
12
[111]
[111]
[111]
½?111?
½?111?
½?111?
½?1?11?
½?1?11?
½?1?11?
½1?11?
½1?11?
½1?11?
ð01?1Þ
ð?101Þ
ð1?10Þ
ð?10?1Þ
ð0?11Þ
(110)
ð0?1?1Þ
(101)
ð?110Þ
ð10?1Þ
(011)
ð?1?10Þ
13
14
15
16
17
18
19
20
21
22
23
24
A
T
A
A
T
A
A
T
A
A
T
A
[111]
[111]
[111]
½?111?
½?111?
½?111?
½?1?11?
½?1?11?
½?1?11?
½1?11?
½1?11?
½1?11?
ð1?21Þ
ð2?1?1Þ
ð11?2Þ
ð?11?2Þ
ð?2?1?1Þ
ð?1?21Þ
ð?121Þ
ð?21?1Þ
ð?1?1?2Þ
ð1?1?2Þ
ð21?1Þ
(121)
Slip systems 1 - 12 do not use labels for twin and anti-twin directions viz. A and T.
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Sa¼ lba
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
X
b
gabðqb
ena:tb
eþqb
sna:tb
sÞ
s
ð5Þ
where l is the shear modulus, gabis the latent hardening coefficient matrix for slip systems a and b. nais the glide plane
normal and tb
in this interaction matrix are chosen so that the hardening behavior generally reflects the salient features of the stress–strain
response reported in the literature. The self interaction coefficient is assumed to be 40% lower (cf. Table 2) than the latent
hardening coefficient. qb
eand tb
sare the tangent directions of edge and screw dislocations on the forest slip system. The coefficients
eand qb
sare the edge and screw type dislocation densities on the forest slip systems.
2.4. Segment length
The average segment length of dislocations laon a mobile slip system a is given by the projection of forest dislocations on
slip system b (Alankar et al., 2011):
la¼
k
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
eand qb
ality coefficient to correlate the average dislocation segment length with the mean free path of the dislocations. For simplic-
ity we assume this coefficient to be identical for edge and screw dislocations.
In this work we use our model approach primarily for performing a parametric study on the effect of different edge and
screw dislocation velocities on giant plastic ansiotropy as described in terms of the Lankford value (also referred to as r-va-
lue). Initial model validation is conducted by the comparison of deformation texture predictions with experimental literature
data. More detailed crystal plasticity simulations of a-Fe single crystals on the basis of the current model (see (Keh, 1965))
and the evolution of the crystallographic texture compared to experimental data will be presented in a subsequent work.
P
bðqb
ena:tb
eþqb
sna:tb
sÞ
q
ð6Þ
where qb
sare the edge and screw type dislocation densities, respectively, on a forest slip system b. k is a proportion-
3. Simulation results and discussion
3.1. Evolution of crystallographic texture
The model described in Section 2 is employed as a user interface subroutine UMAT of ABAQUS (2005). The numerical inte-
gration of the constitutive equations is done as described in the Appendix of the work by Alankar et al. (2009). Initial val-
idation of the model is conducted via simulations of the evolution of the crystallographic texture during plane strain
compression of a-iron. For these simulations a cubic geometry is meshed by using 1728 elements that have 8 nodes and
Table 2
Material constants and parameters for a-Fe used in the current CPFE model.
Elastic constants
C11
C12
C44
|b|
pe
qe
ve;0
F0
kB
se;0
Re
ps
qs
F0,kink
sP
Rs
tD
l0
Self hardening
Latent hardening
k
236.0 GPa
134.0 GPa
119.0 GPa
2.49 e?9 m
0.10
1.11
1.0 e?3 m s?1
3.0 e?19 J atom?1
1.38 e?23 J K?1
20 MPa
19.0 e?9 m
0.71
1.71
2.8e?19 J atom?1
370 MPa
95.0e?9 m
1013s?1
1.0 e?6 m
0.10
0.14
20.0
Burgers vector
Edge dislocation velocity
Screw dislocation velocity
Slip system interaction
Mean-free path coefficient
C11, C12and C44are elastic constants for a-Fe (Adams et al. (2006)). ve;0is the pre-exponential factor in the equation for velocity of edge dislocation. pe and
qe are the exponential coefficients in the equation for velocity of edge dislocations. ps and qs are the exponential coefficients in the equation for velocity of
screw dislocations. F0is the activation energy for edge dislocation motion (Alankar et al., 2011, 2009; Arsenlis and Tang, 2003; Arsenlis and Parks, 2002).
F0,kinkis the activation energy for the motion of screw dislocations via kink pair formation. kBis the Boltzmann constant. se;0is the lattice friction acting on
the motion of edge dislocations. spis the Peierls stress (Stainier et al., 2003) on screw dislocations. l0is the critical kink nucleation length. Ra
critical radii (Arsenlis and Parks, 2002) of interactions for edge and screw dislocations respectively.
eand Ra
sare the
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8 integration points each. Each element is considered as one grain. This setup altogether reproduces a random orientation
distribution and is chosen as starting texture of the initial finite element mesh. As shown in Figs. 2 and 3, the model is able
to predict the typical crystallographic texture evolution of a-iron during plane strain compression for thickness reductions of
70%, 80% and 90% (Hölscher et al., 1991, 1994; Huh et al., 1995; Hutchinson, 1999; Raabe and Lücke, 1993) that can be val-
idated well against the experimental observations as reported in a standard book describing the crystallographic texture of
metals e.g. Kocks et al., (2000).
Fig. 2. Crystallographic texture evolution with 70–90% thickness reduction represented as {001} pole figures. For these simulations only iso-latent
hardening is considered (no dislocation reaction based hardening coefficients). (a), (b), (c) represent the crystallographic texture evolution at 70 %, 80 % and
90 % deformation respectively by using first 12 slip systems in Table 1. (d), (e), (f) represent the crystallographic texture evolution by using all 24 slip
systems in Table 1.
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3.2. Evolution of r-value in electro-deposited pure iron
In this section we apply the constitutive model in order to understand extreme plastic anisotropy effects in terms of the r-
value (Lankford-value) in electrodeposited a-iron sheet. The r-value describes the ratio between the in-plane deformation
component and the deformation component in through-thickness direction. This means that a high r-value describes mate-
Fig. 3. Crystallographic texture evolution (ODFs: orientation distribution functions) with increasing thickness reduction. For these simulations only iso-
latent hardening is considered (i.e. no dislocation reaction based hardening coefficients are considered). (a), (b), (c) represent the crystallographic texture
evolution at 70 %, 80 % and 90 % deformation respectively by using first 12 slip systems in Table 1. (d), (e), (f) represent the crystallographic texture
evolution by using all 24 slip systems in Table 1.
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rial flow primarily parallel to the plane surface in drawing or stretching operation while a low r-value indicates a high por-
tion of normal flow. The former effect (high r-value) indicates good formability in sheet forming operations while the latter
effect (low r-value) indicates rapid sheet thinning leading potentially to premature localization and failure.
As mentioned above, it was observed that electro-deposited pure a-iron having extremely sharp <111>//ND crystallo-
graphic texture shows a very high r-value (ND: normal direction). An experimental assessment reported by Yoshinaga
et al. (2008) shows r-values ranging from 4.8 to 7.1 (see Fig. 5 in Yoshinaga et al. (2008)) for different electro-deposited pure
Fig. 3. (continued)
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iron sheets. These values are much higher than those observed in the best interstitial free IF sheet steels which are charac-
terized by r-values ranging from 1.4 to 2.7 (Yoshinaga et al., 2008). Calculations performed by Yoshinaga et al. (2008) using
different Taylor-Bishop-Hill type isostrain (Taylor Full Constraints model, FC) and relaxed isostrain (Taylor Relaxed Con-
straints model, RC) homogenization models (Van Houtte, 1988) do not predict r-values above 3.2. The FC and RC models that
the authors used for the analysis did not incorporate any details of the microstructure such as dislocation density based
strain hardening or appropriate dislocation velocity laws. As mentioned earlier in Section 2, the crystal plasticity model that
Fig. 3. (continued)
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Fig. 5. Evolution of the predicted r-values as a function of the uniaxial strain in a simulated tensile test for varying sharpness of the initial <111>//ND
texture. In these simulations only multiplication of dislocations was assumed i.e. dislocation annihilation was turned off in order to mimic the effect of the
longitudinal needle shaped grain shapes.
Fig. 4. Various initial orientation distributions used in the simulation study on plastic anisotropy. (a) 100 randomly distributed orientations; c fiber
(<111>//ND) orientations described by (b) 100 orientations with 5? misorientation from the ideal <111>//ND; (c) 100 orientations with 3? misorientation
from the ideal <111>//ND; and (d) 100 orientations with 1? misorientation from the ideal <111>//ND (ND: normal direction).
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we are introducing in this work uses such fundamental microstructure variables as it is based on dislocation mechanisms in
a bcc crystal. One should note though that in the simulations for determining r-values we use only 12 <111>[110] slip sys-
tems like Yoshinaga et al., (2008).
We define the initial crystallographic texture as (a) ‘random’; (b) 5? misorientation from the ideal <111>//ND; (c) 3? mis-
orientation from the ideal <111>//ND (d); and 1? misorientation from the ideal <111>//ND. The pole figures representing
these initial crystallographic textures are shown in Fig. 4.
The orientations shown in Fig. 4 are assigned to a block of 100 elements (8 node brick elements – C3D8 as defined in the
ABAQUSTM(2005) Standard library). For all simulations standard displacement boundary conditions are applied in order to
describe a uniaxial tensile test for a total strain of 10% in tensile direction. Fig. 5 shows the evolution of predicted r-value for
the different initial crystallographic textures as a function of the total tensile strain. Note that for accounting for the effect of
needle shaped highly elongated grains (see Fig. 2 in Yoshinaga et al. (2008)) we turn off the annihilation of dislocations in the
model (i.e. Re= Rs= 0 in Eqs. (3a) and (3b) respectively). The results reveal that with decreasing average misorientation from
the ideal <111> axis, the r-value increases and is extremely large as compared with the one determined for the case with an
initially random texture (r-value ? 1.0). Our simulation results are profoundly different from the FC and RC simulation re-
sults reported by Yoshinaga et al. (2008). Besides the large differences in magnitude of r-values from our simulation results
compared to the values simulated by Yoshinaga et al. (2008) we predict that the r-values are also functions of the gradually
evolving crystallographic texture. This is reasonable as the r-value is a function of crystallographic orientation. Similar to
Yoshinaga et al. (2008), we discuss the r-value at 10% uniaxial tensile strain. For the orientations that have 5?, 3? and 1? aver-
age misorientation, respectively, from the ideal <111>//ND texture, the r-values are ?5.6, 7.2, and 7.6 respectively after 10%
strain. If the annihilation of the dislocations is not turned off in the constitutive description i.e. Reand Rsare not set to zero in
Eqs. (3a) and (3b), respectively, than the r-value does not become so high. For example, for 1? misorientation from the
<111>//ND texture, the r-value decreases from ?7.6 to ?6.5 at 10% strain in this case. These results indicate that in large
grains in which dislocations can substantially annihilate before reaching grain boundaries, the r-value will not be as high
as in long needle-shaped grains.
Further, we explain this unexpected phenomenon of large r-values in terms of dislocation loop activities in the needle
shaped grains. As mentioned earlier, in a-Fe, edge dislocations move much faster than screw dislocations. Owing to this ki-
netic constraint, the best possible accommodating expansion of such dislocation loops should be the one in which screw dis-
location segments are oriented along the larger dimension of the needle shaped grains. This arrangement allows the
maximum unrestricted glide of edge and screw dislocations according to their velocities. Such shape of dislocation loops
arises necessarily when accounting for the different glide rates of the two types of dislocations. Fig. 6 schematically shows
one of the long grains having strong <111>//ND texture formed during the electrodeposition of a-Fe (Yoshinaga et al., 2008).
Formability is interpreted as the ability of a metal to undergo severe shape changes without fracture. In terms of dislocation
theory, better formability means the ability to deform without redundant interaction of dislocations. In the present case, the
redundant interaction of dislocations is bypassed due to the fact that accommodating glide paths available to the edge and
screw dislocations, as prescribed by the grain shapes, are matching their natural expansion in accord with their individual
velocities. Therefore such a combination of dislocation velocities (? ma
grain shapes consequently enables enhanced formability. Specifically under these crystallographic and topological boundary
conditions the alignment of the screw dislocations is parallel to the longitudinal axis of the needle shaped grains. One should
note that for bcc metals we essentially do not need to turn off the annihilation of screw dislocations since their velocity is
already so low that they will not undergo substantial annihilation. For achieving a similar case for fcc crystals it is necessary
to (artificially) turn off the annihilation of both edge and screw dislocations in the model.
e>? ma
s), crystallographic <111> texture, and elongated
Fig. 6. A dislocation loop expanding under the effect of applied stress in one of the grains oriented such that <111>//ND. In such grains, it is highly probable
that most of the screw dislocation lines are aligned along the ND with slight misorientation.
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Fig. 7. Slip system activity in the simulations with (a) 1? misorientation from the ideal <111>//ND; (b) 3? misorientation from the ideal <111>//ND; and (c)
5? misorientation from the ideal <111>//ND fiber orientation (ND: normal direction). The numbers 1;2;3... indicate the individual slip systems according
to Table 1.
A. Alankar et al./International Journal of Plasticity 52 (2014) 18–32
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The physical interpretation of these simulation results is given in terms of the underlying slip systems and their kinetics.
Fig. 6 shows a dislocation loop expanding under the effect of the applied stress in one of the grains oriented such that the
crystal <111> direction aligns with the normal direction of the tensile sample (ND). In such grains, it is highly probable that
most of the screw dislocation lines are aligned along ND with the slight misorientation scatter used as a parameter as de-
scribed above. By decreasing the misorientation from <111>//ND it is ensured that the slip systems available in the grains
are the ones which have 111-screw dislocations along the length of the grains. One should note that the grains have a strong
<111>//ND texture so there is very small probability of other slip systems to become critically stressed. Our simulations
show that in most of the grains ?4 slip systems are active in such a case and this number of slip systems decreases as
the misorientation relative to the <111> axis decreases (cf. Fig. 7). This effect decreases the amount of latent hardening
and most of the hardening affecting the deformation is due to the interaction of dislocations with grain boundaries after
a long and undisturbed mean free glide path. The interaction of grain boundaries with these dislocations is not included
in the constitutive model though, but the effect of different mobilities of edge and screw dislocations on the formability
is obvious. Note in Fig. 7, that the dislocation density evolution rate and total dislocation density at 10% tensile strain is very
high since for replicating the effect of needle shaped grains we do not account for annihilation of dislocations (Re= Rs= 0 in
Eqs. (3a) and (3b)). With an increase in the misorientation of the starting texture away from the ideal <111> ND axis, more
and more slip systems become active and higher strain hardening takes place. The increase in strain hardening in turn is
reflected by the decreased r-value for those simulations that start with larger average misorientations of the grains from
the ideal <111> axis.
In the present case, the orientation of the screw dislocations is such that there is very low resolved shear stress on the
screw dislocations. This low available shear stress and high Peierls stress make the movement and therefore annihilation
of screw dislocations very difficult. The effect of annihilation of edge dislocation segments will not be strong since they
are much smaller in length. However, due to high velocity of edge dislocation segments, the length of screw dislocation seg-
ments will increase quickly. This effect causes more multiplication of dislocations than the effect of annihilation if any. A
similar effect of the annihilation of dislocations is shown in Fig. 8. The lowest r-values are achieved when regular annihila-
tion of dislocations is taking place (Re¼ Rs–0 in Eqs. (3a) and (3b)). In Fig. 8, we study the effect of the individual dislocation
velocities and the different conditions of annihilation of the dislocations for grain orientations which are oriented within 5?
misorientation from <111>//ND. Interestingly, for a 5? spread of the orientations away from the ideal <111>//ND texture,
the highest r-value is seen when the annihilation of dislocations is (artificially) turned off (Re= Rs= 0 in Eqs. (3a) and (3b)
respectively) and the velocities of the edge and screw dislocations are identical ve=vs. This constitutive rule mimics a topo-
logical situation of long needle shaped grains for bcc crystals being deformed at high temperature where no friction barrier
acts against the free motion of screw dislocations (Seeger, 2001) and dislocation annihilation is not taking place. Note that if
Reand Rsare not set to zero then the r-value is not that high (Reand Rsindicate the critical interaction radii for edge and
screw type dislocations respectively). This result is shown by the pink curve in Fig. 8.
Finally, the overall effect on r-value can be described as a type of grain topology or, more specific, grain-shape effect. In
the present case it is shown that this effect is built-in into the microstructure and is an outcome of the dislocation mean free
path and the different velocities of edge and screw dislocations in a bcc crystal. The r-values that we predict with this ap-
proach are in good agreement with the experimental data (4.1–7.0) (Yoshinaga et al., 2008). From our approach we can spe-
cifically learn that such macroscopic quantities as the Lankford anisotropy factor (r-value) can depend to a large degree not
Fig. 8. Effect of dislocation velocity and dislocation density evolution on the evolution of the r-value. Reand Rsrepresent the critical interaction radii for
edge and screw dislocations respectively as defined in Eq. (3). veand vsare velocities of edge and screw dislocations respectively.
30
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only on the crystallographic texture but also on the underlying dislocation kinetics. This is reflected by Yoshinaga’s work
(Yoshinaga et al., 2008) who suggested that the observed giant planar anisotropy could not be modeled by using traditional
micromechanical homogenization approaches (FC, RC Taylor models) alone (Van Houtte, 1988) although they served well
over several decades for describing certain features of the crystallographic texture evolution. However, instead the authors
suggested that more flexible constitutive models should be developed which would be then able, for instance, to predict the
high r-values in electro-deposited pure iron sheets having long needle shaped grains. Delannay and Barnett (2011), recently,
have attempted to model such plastic anisotropy based on explicit modeling of the grain size and grain shape by using a
phenomenological crystal plasticity model.
4. Conclusions
We present a physically based crystal plasticity model for finite deformation of bcc crystals. The novelty of the constitu-
tive approach is that it considers the individual kinetics of screw and edge dislocations that can lead to the formation of char-
acteristic dislocation loops in a bcc crystal. The constitutive framework is self sustained and does not require the commonly
used power law formulations. More specifically the model distinguishes between the edge and screw dislocations via differ-
ent velocities of the two types of dislocations in a bcc crystal. The velocity of the screw dislocations is accounted by the for-
mation of kink-pairs. The model is able to predict the evolution of the crystallographic texture reasonably well. We use the
model for predicting the excellent formability in terms of the r-value of electro-deposited pure iron which has a strong
<111>//ND texture. The model shows excellent predictions of the r-value as compared with experiments. The r-value in-
creases with decreasing misorientation of the average grain orientation from the ideal <111>//ND axis. This effect cannot
be so well predicted by the model without accounting for different velocities of edge and screw type dislocations and also
not without turning off the annihilation of the dislocations (due to low velocity of screw dislocations). A major advantage of
the models like the one presented here is that their modular nature enables us to include also further deformation mecha-
nism at the generic dislocation level. The work gives an example of the fact that models that are based on fundamental dis-
location mechanisms in crystal plasticity, are suited not only for understanding the underlying physics at the single
dislocation level but they also can replace more traditional homogenization approaches that rely on phenomenological
power-law formulations.
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