Materials Transactions, Vol. 56, No. 7 (2015) pp. 963 to 972
Special Issue on Long-Period Stacking Ordered Structure and Its Related Materials II
© 2015 The Japan Institute of Metals and Materials
Crystal Plasticity Analysis of Development of Intragranular Misorientations due
to Kinking in HCP Single Crystals Subjected to Uniaxial Compressive Loading
Tsuyoshi Mayama1,+, Tetsuya Ohashi2, Yuichi Tadano3 and Koji Hagihara4
1
Department of Materials Science and Engineering, Kumamoto University, Kumamoto 860-8555, Japan
Mechanical Engineering, Kitami Institute of Technology, Kitami 090-8507, Japan
3
Department of Mechanical Engineering, Saga University, Saga 840-8502, Japan
4
Department of Adaptive Machine Systems, Osaka University, Suita 565-0871, Japan
2
The mechanism and the effective factors for the development of intragranular misorientations due to kinking is studied by a crystal
plasticity finite element method. A single crystal with hexagonal close-packed (HCP) structure in which only basal slip system is activated is
used as a model material. To activate basal slip system, the initial crystal orientations are set to be the ones whose basal planes are slightly
deviated from the compressive direction. The result shows that basal slip and the development of intragranular misorientations are sometimes
localized near the center of the specimen depending on the initial deviation angle, strain hardening rate, and strain rate sensitivity. The
mechanism is discussed in terms of the nonuniform stress distribution and lattice rotation. The effect of slight changes in the boundary conditions
shows significant effect on the positions of slip localization. In summary, the present numerical results suggest that there are a number of
effective factors for the development of the intragranular misorientations due to kinking including initial crystal orientation, material parameters,
and boundary conditions. [doi:10.2320/matertrans.MH201403]
(Received December 25, 2014; Accepted February 13, 2015; Published March 27, 2015)
Keywords: crystal plasticity, finite element analysis, kinking, single crystal
1.
Introduction
The electron backscatter diffraction (EBSD) technique has
been used widely to measure intragranular misorientations in
crystal grains and detailed features of microstructure in
deformed metals were examined.1­4) Pantleon et al.1) characterized the anisotropy of orientation distributions within
individual grains in cold-rolled aluminium. Scheriau and
Pippan2) reported the influence of grain size on orientation
change in polycrystalline copper, nickel and iron. The
orientation gradient around a hard particle in a warm-rolled
Fe3Al-based alloy3) and the distribution of geometrically
necessary dislocations in dual-phase steels4) were studied by
three-dimensional EBSD. These results suggest that intragranular misorientation is a key indicator for material
properties and microstructure evolution. However, the detailed mechanism and effective factors in the development of
intragranular misorientation has not been clarified.
Intragranular misorientations evolve when the plastic slip
deformation is spatially nonuniform and different amount of
lattice rotation take place. While in polycrystals the mutual
interactions between grains lead to inhomogeneous deformation, numerical studies on single crystals5­10) show that
deformation can also be inhomogeneous depending on
boundary conditions and initial fluctuations. Therefore,
inhomogeneous deformation of single crystals is considered
to be an essential point for the understanding of the
development mechanism of intragranular misorientations.
Among several deformation modes that lead to inhomegeneous deformation of single crystals, kinking often induces
large intragranular misorientations.9,10) While kinking is one
of common deformation modes in metals,11­16) intensive
studies have been made on hexagonal close-packed (HCP)
crystals where basal slip system was dominantly activated.17­24) Orowan considered kinking as third fundamental
+
Corresponding author, E-mail: [email protected]
deformation mechanism as well as slip and twinning.17) In
contrast, Hess and Barrett18) made a careful observation of
compressed Zn single crystals and proposed a simple model
of kink formation by an array of dislocations. Jillson19) also
concluded that kinks developed after gliding or slip processes
from experimental results obtained for Zn single crystals.
Hess and Barrett18) discussed that the positions of kink
formation varied depending on boundary conditions and
defects. Meanwhile, Gilman22) showed that kinks tended to
form near the center of specimen, which disagreed with the
results by Hess and Barrett.18) Gilman22) also showed that the
formation of kink depended on crystal orientation.
Whereas these intensive experimental researches, it is still
difficult to specify which are the dominant factors for
development of intragranular misorientations because the
experimental results suffer influences from various origins
and unavoidable initial imperfections. While kinking is
mentioned in several theoretical and/or numerical researches,8­10,25) detailed investigation of the influences of
various parameters on the development of intragranular
misorientations has not been reported.
In this study, we try to gain fundamental understandings of
the development of intragranular misorientation due to
kinking. We employ HCP crystals for a model material in
which only basal slip system is activated. A crystal plasticity
finite element method is used to evaluate the effects of initial
crystal orientation, strain rate sensitivity and strain hardening
rate on the development of intragranular misorientations and
kinking. Mechanism of the development of intragranular
misorientations, abrapt drops in the stress-strain response
during kinking and the influence of slight change in boundary
conditions are discussed.
2.
Overview of the Intragranular Misorientations Due
to Kinking
Figure 1 shows the typical kink bands experimentally
964
T. Mayama, T. Ohashi, Y. Tadano and K. Hagihara
(a)
(b)
1010
-2110
0001
(d)
(e)
0001
-2110
(c)
25
(h)
(g)
L1
L2
100 μm
20 μm
Misorientation from
the origin, degrees (°)
(f)
20
L1
15
10
L2
5
0
0
5
10
15
Distance from the origin, l/μm
Fig. 1 Typical kink bands experimentally observed in Zn single crystal and polycrystalline Mg-based 18R LPSO phase. (a) Shape of
direction at room temperature. The observed direction of the
deformed Zn single crystal after 20% compressive strain along the ½0110
(b) Crystal orientation map examined by SEM-EBSD pattern analysis in specimen after 3% compressive
specimen is parallel to ½21 10.
and (0001) pole
strain. The boundaries across which the lattice is rotated more than 5 degree are drawn by black lines. (c)­(e) f21 10g
figures taken at the positions marked A­C in Fig. 1(b), respectively. (f ) Surface morphology of directionally solidified Mg-based 18R
LPSO phase alloy after compressive loading along nearly parallel to basal plane. (g) Crystal orientation map in the deformed Mg-based
18R LPSO phase alloy. (h) Line profiles indicating the variations in crystal orientation along L1 and L2 in Fig. 1(g).
observed in Zn single crystal and directionally-solidified
(abbreviated as DS, hereafter) polycrystalline Mg-based 18R
LPSO (long period stacking ordered) phase alloy with the
composition of Mg85Zn6Y9 (at%).26) Figure 1(a) shows an
optical micrograph of kink bands in a Zn single crystal
specimen compressed 20% plastic strain along the ½0110
direction. The observation direction was parallel to the
direction. A significant change in the shape of the
½21 10
specimen is observed. Black lines in Fig. 1(a) correspond to
the trace of basal planes those were introduced by chemical
etching. They are sharply bent at localized regions near the
top and bottom ends and kink bands accompany with these
lattice rotations.
The microstructures in the Zn single crystal specimen in
the early stages of deformation were examined by electron
backscatter diffraction (EBSD) pattern analysis with scanning
electron microscopy (SEM). Figure 1(b) shows the crystal
orientation map after 3% compressive strain. The observation
direction. Boundaries
direction was parallel to the ½21 10
across which lattice rotation was larger than 5° were drawn
by black lines. A wide kink band introduced by deformation
appeared in the center of the image shown in Fig. 1(b).
and (0001) pole figures taken
Figures 1(c­e) display f21 10g
at positions A­C in Fig. 1(b), respectively. These demonstrate that a large crystal rotation was caused by the formation
axis that was perpendicular
of a kink band along the ½21 10
to the normal to the (0001) basal slip plane. Crystal rotations
across the sites A and B, and B and C are about 34° and 32°,
respectively. Note that these lattice rotation angles are not
constant and the value is different in each deformation band.
Moreover, some boundaries terminate within the crystal, as
indicated by arrows in Fig. 1(b).
Although the basal slip is the easiest deformation mode in
the Zn single crystal, the contributions of non-basal slip and
twinning cannot be completely avoided. While it was
recently reported that in the Mg-based LPSO phase DS
crystal only basal slip system was predominantly activated
compared to the other slip systems in deformation at room
temperature.26,27) Because of this fact, contributions from
non-basal slips and twinning to kinking should be limited in
the case of this alloy system. Figures 1(f ) and (g) shows an
optical micrograph of the surface morphology and the crystal
orientation map in the deformed LPSO phase DS crystal. The
specimen was compressed at room temperature along the
growth direction where the basal plane was almost parallel to
the loading axis in most of the grains (vertical direction in
the Fig. 1(g)). A number of sharp bends of basal plane are
observed and these are considered to be developed by
superimpose of smaller and ridge-shaped kink bands.
Figure 1(h) shows the line profiles of misorientation angle
from the origin along L1 and L2 indicated in Fig. 1(g).
Although there are significant changes in crystal orientation
near the both kinked regions, the transitions of crystal
orientation along L1 and L2 are quite different. The
transitions along L1 and L2 are step-like and triangular,
respectively.
Crystal Plasticity Analysis of Development of Intragranular Misorientations due to Kinking in HCP Single Crystals
In these ways, experimentally observed deformation
microstructure by kinking are considerably complicated.
In the LPSO phase polycrytalline specimen, the effect of
interactions between grains cannot be excluded. Additionally, in any experimental studies, effects of frictions on
loading surfaces and initial imperfections are usually
contained. To exclude such effects to kinking and to clarify
the formation mechanism of deformation microstructure,
numerical analysis using a crystal plasticity finite element
method is useful. In the present study, kink band formation
behavior was examined by using HCP single crystals with
no initial imperfections those deform by only basal slip
system.
3.
(a)
(b)
θ ini
c
z
z
y
x
(c)
θ ini
c
BSL1
c
y
BSL3
BSL2
z
θ ini
x
BSL3
BSL2
BSL1
(d)
z
y
x
BSL3
965
BSL2
z
z
BSL1
15°
z
x
x
y
x
y
x
y
y
Fig. 2 Schematic diagrams of analysis model. (a) Model geometry and
10,
(c) ½0110
and (d) intermediate
boundary conditions. For near (b) ½12
compressive loadings, the definitions of initial deviation angle ªini and
deformation modes.
Numerical Procedure
3.1 Crystal plasticity finite element method
In this study, the rate-dependent finite strain crystal
plasticity model proposed by Peirce et al.6) was used. The
crystal plasticity model was implemented into each gauss
point in a large-deformation finite element scheme as a
constitutive law and the development of heterogeneous
distributions of slip accumulation and crystal orientation
were calculated.
After the proposal of the rate-dependent finite strain crystal
plasticity model, considerable developments in crystal
plasticity model have been made with implementation of
geometrically necessary dislocation (GND) density based on
strain gradient.28­30)
When the kink deformation takes place, GND density
should significantly increase with the development of sharp
bending of slip plane. At the same time, back stresses31­33)
and additional dissipations due to the development of strain
gradient or the evolution of plastic curvature and torsion33,34)
and other factors associated with GNDs will be important
issues to be discussed. However, thorough discussions of
these effects are beyond the objective of this study. Instead,
classical crystal plasticity model is used to gain primary
understandings of the effects including the initial crystal
orientation, strain hardening rate, rate sensitivity and
boundary conditions in kinking.
In the present paper, the shear slip rate £_ ðiÞ for deformation
mode i was calculated by
ðiÞ 1=m
¸ ð1Þ
£_ ðiÞ ¼ £_ 0 sgnð¸ ðiÞ Þ ðiÞ :
g
Here, sgnðxÞ ¼ 1 if x 0 and sgnðxÞ ¼ 1 if x < 0; £_ 0 and
m are the reference shear strain rate and the strain rate
sensitivity parameter, respectively. The ¸ ðiÞ and gðiÞ are the
resolved shear stress (RSS) and reference stress for
deformation mode i, respectively. The following equation
was used for gðiÞ in this study:
X
g_ ðiÞ ¼
hðijÞ j£_ ðjÞ j
ð2Þ
j
ðijÞ
where h is the strain hardening rate. A constant value of h
was used for hðijÞ for simplicity, although the strain hardening
rate should depend, in general, on various parameters,
including the loading history, the reactions of dislocations,
temperature, and so forth.
For numerical integration of the constitutive model, the
rate-tangent modulus method proposed by Peirce et al.6) was
used. To ensure stable calculations even when the slip is
localized, an r-minimum strategy35,36) was employed to
control the increment of time.
3.2 Analysis model
Figure 2 shows the schematic diagrams of analysis model.
The model is a square prism with dimensions of 10 ©
10 © 20 µm. The model is uniformly divided into 10 ©
10 © 20 finite elements of 20-node solid type. Reduced
integration is adopted and the number of integration points
for each element is eight. The boundary conditions are
illustrated in Fig. 2(a). The displacement in the z-direction
at the bottom surface is fixed and a uniform compressive
displacement in the z-direction is applied at the top surface.
The displacements in the x- and y-directions at the top and
bottom surfaces are not fixed; this corresponds to a condition
under which there is no friction between the compressive jig
and the specimen. The lateral surfaces of the model are
traction-free.
To induce a single slip of the basal slip system, the loading
10
direction. When the
direction was set to be near the ½12
loading direction is exactly the ½1210 direction, the basal slip
system is not activated because the Schmid factors for the
basal slip system become zero. To make the Schmid factors
of the basal slip system nonzero, the loading direction should
10
direction. The initial deviation angle
deviate from the ½12
ªini, defined as the angle between the loading direction and
the basal plane as shown in Fig. 2(b), was introduced in the
model. That is, the initial crystal orientation was set so that it
10
direction by a rotation of ªini about
deviated from the ½12
the x-axis (½1010 axis). For every Gauss point, the same
initial crystal orientation was allocated. In Fig. 2(b), the
definitions of the deformation modes are also shown. Here,
the three modes of basal slip systems are named as BSL1,
BSL2 and BSL3. The effects of the initial deviation angle ªini
and the finite element mesh are discussed in Sec. 4.1. In
Sec. 4.5, effects of slight changes in the constraints on the top
or/and bottom surfaces are discussed.
Deformation of specimens where the compressive direc direction and
tions were slightly deviated from the ½0110
were also
intermediate direction between ½1210 and ½0110
made in Sec. 4.6. The initial orientations and deformation
modes are schematically illustrated in Figs. 2(c) and (d).
966
T. Mayama, T. Ohashi, Y. Tadano and K. Hagihara
Table 1 Model parameters used in crystal plasticity analysis.
Section
(Fig.)
g(basal)
[MPa]
E
[GPa]
¯
ªini
4.1
(Figs. 3, 4, 6 and 7)
45
0.35
3, 4 or 5
100
0.02
10 © 10 © 20
10 © 10 © 20
10
4.2
(Fig. 8)
45
0.35
3
50, 100, 200 or
300
0.02
10 © 10 © 20
10 © 10 © 20
10
4.3
(Fig. 9)
45
0.35
3
100
0.01, 0.02, 0.05 or
0.1
10 © 10 © 20
10 © 10 © 20
10
4.4
(Fig. 10)
45
0.35
3
100
0.02
10 © 10 © 10,
10 © 10 © 20 or
10 © 10 © 40*1
10 © 10 © 10,
10 © 10 © 20 or
10 © 10 © 40*1
10
4.5, 4.6 and 5.1
(Figs. 4*2, 11, 12 and 13*3)
45
0.35
3
100
0.02
10 © 10 © 20
10 © 10 © 20
10
h
[MPa]
Mesh discretization
[element]
m
Mesh geometry
[µm]
*1The initial size of each element for all models is 1 © 1 © 1 µm.
*2Lattice rotation is suppressed in gray dashed-dotted line in Fig. 4.
*3The initial crystal orientations of the results in Figs. 13(b) and (c) are different from the others by rotating about c-axis as shown in Figs. 2(c) and (d).
4.
Numerical Results
4.1 Effect of initial deviation angle
Figure 3 shows the development of intragranular misorientations during compressive loading. To visualize the
variation of the crystal orientation, the deviation angle ¦ª
from ªini is defined as shown in Fig. 3(a). Here, a positive
value of ¦ª indicates an increase in the deviation angle
between the loading direction and basal plane. In Fig. 3(a),
distributions of ¦ª at nominal strains ¾zz = ¹0.5, ¹0.6
and ¹0.7% in the model with ªini = 3° are illustrated. In
Fig. 3(a), ¦ª continuously increases near the center of the
model. Figure 3(b) shows the relationship between the
maximum ¦ª at each strain and the nominal compressive
strain. The model with ªini = 3° show rapid increase in ¦ª
around compressive strain of 0.5%. In contrast, the change in
¦ª in the model with ªini = 5° is much more gradual.
Figure 4 shows the calculated stress-strain curves of single
crystals with ªini of 3, 4 and 5°. The nominal stress-strain
curves differ significantly with the initial crystal orientation;
that is, when ªini = 3°, the stress rapidly decreases during
deformation, whereas the stress gradually decreases for
ªini = 5°. The smaller the initial deviation angle ªini, the
larger the amount and the rate of the decrease in stress. The
peak stress immediately before the decrease in stress also
(a)
(b) 20
z
θini = 3°
Maximum Δθ (°)
θ ini+ Δθ
z
εzz = -0.5% εzz = -0.6% εzz = -0.7%
x
y
Δθ [°] : 0
13.5
15
θini = 4°
θini = 5°
10
5
0
0
0.2
0.4
0.6
0.8
1
Nominal compressive strain (%)
Fig. 3 Change in crystal orientation during compressive loading. (a) Distributions of ¦ª in the model with ªini = 3°. (b) Changes in the maximum
¦ª at each strain in the model with ªini of 3, 4 and 5°.
Nominal compressive stress, σ / MPa
3.3 Deformation mode and material parameters
basal slip is the sole mode of plastic
The ð0001Þh1120i
deformation considered in the present analysis. The initial
value of the reference stress gðiÞ for the basal slip system is set
to be 10 MPa. Young’s modulus E, Poisson’s ratio ¯, rate
sensitivity factor m and strain hardening rate h were set to be
the experimentally reported values for conventional polycrystalline Mg alloys at room temperature.37,38) The effects of
the variation in strain hardening rate and strain rate sensitivity
are discussed in Sec. 4.2 and 4.3, respectively. Apparent
causes of non-uniform deformation such as initial fluctuations of material parameters, geometrical imperfection are not
included in this paper. Parameters used in the calculations in
each section are summarized in Table 1.
350
Without lattice
rotation (θ ini = 3°)
300
250
B
200
A
C
150
100
θ ini = 3°
θ ini = 4°
θ ini = 5°
50
0
0
0.2
0.4
0.6
0.8
1
Nominal compressive strain (%)
Fig. 4 Variations in calculated stress-strain curves with the initial deviation
angle ªini of 3, 4 and 5°.
depends on the initial crystal orientation. The different peak
stresses are the result of different initial Schmid factors,
which in turn depend on the crystal orientation. It was
confirmed that the difference in the maximum RSS values at
the peak stresses in single crystals with ªini of 3, 4 and 5 is
within 1%.
The comparison between Fig. 3(b) and Fig. 4 indicate that
the significant decrease in stress for smaller ªini is associated
with the rapid and pronounced change in crystal orientation.
The mechanism of the decrease in stress and localization is
discussed in Sec. 5.1.
250
(a)
967
(b)
25
z
200
Misorientation Δθ (°)
Nominal compressive stress, σ / MPa
Crystal Plasticity Analysis of Development of Intragranular Misorientations due to Kinking in HCP Single Crystals
150
100
10 × 10 × 10 elements
10 × 10 × 20 elements
50
10 × 10 × 10
elements
20
15
10 × 10 × 20
elements
l
10 × 10 × 40
elements
y
10
x
5
10 × 10 × 40 elements
0
0
0.2
0.4
0.6
0.8
0
1
0
0.2
0.4
0.6
0.8
1
Normalized position in z-direction x3 / l
Nominal compressive strain (%)
Fig. 5 Effect of finite element mesh resolution on the macroscopic stress-strain curves and on the distribution of ¦ª in the model with
ªini = 3°. (a) Stress-strain curves. (b) Line profiles of ¦ª at ¾zz = ¹1%.
z
x
θ ini = 3°
y
θ ini = 5°
θ ini = 4°
γ acc : 0
0.36
Fig. 6 Distribution of the accumulation of shear slip of the basal slip
system, BSL2, at ¾zz = ¹1% in models with ªini = 3, 4 and 5°.
(a)
25
(b)
z
x
θ ini = 3°
y
θ ini = 4°
Δθ [°] : 0
θ ini = 5°
Misorientation Δθ (°)
As is well known, the calculations of localization by the
classical finite element method show mesh dependence.39)
Figures 5(a) and (b) show the effects of the mesh resolution
on nominal stress-strain curves and the change in the crystal
orientation when the nominal compressive strain is ¾zz =
¹1% in the model with ªini = 3°. In Fig. 5(b), the line
profiles of ¦ª at the center of x-y plane along z-direction are
compared. Here, normalized position in z-direction is defined
by x3 (coordinate value in z) divided by model length l. The
line profiles show triangular transition of crystal orientation
which are similar to the experimental result of L2 in
Fig. 1(h).
The results for meshes uniformly divided into 10 ©
10 © 10, 10 © 10 © 20 and 10 © 10 © 40 elements, in which
the geometry of the analysis model is fixed to 10 © 10 ©
20 µm, are compared. Although the stress-strain curves are
almost the same, as shown in Fig. 5(a), the line profile of ¦ª
clearly depends on the mesh resolution. The finer meshes
exhibit more significant variation of ¦ª near the center of the
model as shown in Fig. 5(b). Additionally, the result of the
finer mesh shows that the variation of ¦ª is localized in the
narrower width compared with the results of the coarser
mesh.
As shown above, the results of localization clearly depend
on the finite element mesh resolution. Therefore, careful
attention should be paid on finite element mesh when
quantitative evaluation. In the present study, the influences of
parameters are examined by using the same finite element
mesh. As discussed in literatures,32) higher order theory of
plasticity could be possible to avoid such mesh dependency.
In the all results as shown above, the accumulations of
shear slip by BSL1 and BSL3 are less than 1% of that for
BSL2 at a nominal compressive strain of ¾zz = ¹1%. That is,
the single-crystal model is deformed by the approximately
single slip of BSL2. Figure 6 shows the effect of ªini on the
distribution of the accumulation of shear slip of basal slip
system BSL2 when the nominal compressive strain is ¾zz =
¹1%. The distribution strongly depends on ªini. In the models
with smaller ªini, more basal slip is accumulated near the
center of the model. With the localized accumulation of basal
slip, the model becomes bent.
Figure 7 shows the dependence of ªini on the variation of
crystal orientation when the nominal compressive strain is
¾zz = ¹1%. Figure 7(a) shows the distributions of ¦ª in the
θ ini = 3°
20
θini = 4°
θ ini = 5°
15
10
5
18
0
0
0.2
0.4
0.6
0.8
1
Normalized position in z-direction x3 / l
Fig. 7 Effect of ªini on development of intragranular misorientation.
(a) Distributions of ¦ª at ¾zz = ¹1% in the model with ªini of 3, 4, and
5°. (b) Line profiles of ¦ª at ¾zz = ¹1%.
models with ªini of 3, 4 and 5°. The distributions clearly
indicate that more pronounced intragranular misorientations
are introduced in the models with smaller ªini. In the center
region of the model with ªini = 3°, the value of ¦ª varies
significantly. Comparing Fig. 6 with Fig. 7(a), it is found that
the distributions of the accumulations of basal slip and ¦ª are
qualitatively very similar. This similarity arises as a result of
lattice rotation caused by basal slip as discussed in Sec. 5.1.
Figure 7(b) show the line profiles of ¦ª at the nominal
compressive strain is ¾zz = ¹1%. With decrease in ªini,
significant misorientation is introduced at the narrow region
near the center especially in the case of ªini = 3 and 4°. On
the other hand, the model with ªini = 5° shows the larger
change in the ¦ª near top and bottom regions compared with
those in the model with ªini = 3 and 4°.
The axis of lattice rotation in all models in Fig. 7 is the x
axis (½1010
axis) as explained in Sec. 5.1. The lattice
rotation about the x-axis leads to the bending of basal planes.
T. Mayama, T. Ohashi, Y. Tadano and K. Hagihara
300
25
(a)
(b)
h = 300 MPa
250
Misorientation Δθ (°)
Nominal compressive stress, σ / MPa
968
h = 200 MPa
200
150
h = 100 MPa
100
h = 50 MPa
50
0
0
0.2
0.4
0.6
0.8
Nominal compressive strain (%)
1
h = 50 MPa
20
h = 100 MPa
h = 200 MPa
15
h = 300 MPa
10
5
0
0
0.2
0.4
0.6
0.8
1
Normalized position in z-direction x3 / l
Fig. 8 Effect of strain hardening rate of basal slip on macroscopic stress-strain curves and on the distribution of ¦ª in the model with
ªini = 3°. (a) Variation in stress-strain curves. (b) Line profiles of ¦ª at ¾zz = ¹1%.
The intragranular misorientations revealed in this calculation
can be interpreted as kink bands because the boundaries
of intragranular misorientations are not parallel to the slip
direction of the active slip plane. In other words, the
boundaries should be ideally parallel to shear direction when
the boundaries are formed by slip banding or twinning.
4.2 Effect of strain hardening rate
In the previous section, the calculations were performed
with a strain hardening rate h of 100 MPa. In this section, the
effect of the strain hardening rate on the distribution of
intragranular misorientations is discussed. While experiments
on HCP single crystals indicate a very low strain hardening
rate of the basal slip system, the strain hardening rate
significantly increases with the introduction of a slight lowangle boundary into single crystals.40) Therefore, it is
valuable to study the effect of the strain hardening rate on
the development of intragranular misorientations.
Figure 8(a) shows the effect of the strain hardening rate on
the nominal stress-strain curve in the model with ªini = 3°.
Here, the strain hardening parameter h is set to 50, 100, 200,
or 300 MPa. Stress-strain curves for h smaller than 300 MPa
show a decrease in the flow stress from the peak stress,
whereas the stress continuously increases when h = 300
MPa. Figure 8(b) shows the line profiles of ¦ª at a nominal
compressive strain of ¾zz = ¹1% for different values of h.
Significant intragranular misorientations are developed near
the center of the model when h is 50 or 100 MPa. In contrast,
the difference in ¦ª is slight except near the top and the
bottom in the model when h is 200 or 300 MPa.
4.3 Effect of strain rate sensitivity
In eq. (1), m is the strain rate sensitivity parameter. The
strain rate sensitivity is dependent on the material, microstructure, temperature, slip system, and so forth. Even among
HCP metals, strain rate sensitivities are significantly different.
While beryllium and zinc show significant strain rate
dependence even at room temperature,41,42) the strain rate
sensitivity of magnesium at room temperature is relatively
low.43) In this section, to clarify the effect of strain rate
sensitivity on the development of intragranular misorientations, the numerical results for four different strain rate
sensitivities are compared.
Figure 9(a) shows the effect of the strain rate sensitivity on
the nominal stress-strain curve in the model with ªini = 3°.
Here, a strain rate sensitivity parameter m of 0.02, 0.05, 0.1
or 0.2 is used. The rate of the decrease in stress from the peak
stress decreases with increasing m. This trend is qualitatively
consistent with the numerical results reported by Peirce
et al.30) Figure 9(b) shows the line profiles of ¦ª at a
nominal compressive strain of ¾zz = ¹1%. A smaller m leads
to significant intragranular misorientations while the results
with larger m show the smaller difference in ¦ª.
As in a number of previous researches18­22) and in the
Sec. 2, Zn single crystals frequently show significant kinking
even though the strain rate sensitivity at room temperature is
relatively large. One possible reason for the frequent kinking
in Zn would be the high ratio of lattice constants c/a. Due to
twinning is geometrically prevented
the high ratio, f1012g
under the loading in the parallel direction to basal plane.44)
In that case, kinking should accommodate the compressive
strain in the parallel to basal plane in place of f1012g
twinning. The other possible reasons are the influences of
several effective factors including strain hardening rate and
boundary conditions as discussed in the present study.
However, to discuss the detailed mechanism of kinking in
Zn is beyond the scope of the present study because the
activity of non-basal slip system and the interaction between
basal and non-basal slip systems cannot be ignored for Zn
even at room temperature.
4.4 Effect of aspect ratio
The effect of aspect ratio of the analysis model is
examined. Figures 10(a) and (b) show the effects of the
aspect ratio on the nominal stress-strain curves and the line
profiles of ¦ª at a nominal compressive strain of ¾zz = ¹1%
in the model with ªini = 3°. In the calculations, models with
geometries of 10 © 10 © 10, 10 © 10 © 20 and 10 © 10 ©
40 µm are used. These models are uniformly divided into
10 © 10 © 10, 10 © 10 © 20 and 10 © 10 © 40 elements
with the same initial size of each element of 1 © 1 © 1 µm.
Although the stress-strain curves are almost the same, as
shown in Fig. 10(a), the line profiles of ¦ª show a
dependence on the model geometry as shown in Fig. 10(b).
The model with the higher aspect ratio exhibits more
significant intragranular misorientations near the center of
the model. The misorientation is more localized in the
narrower region in the result with higher aspect ratio
compared with the result with lower aspect ratio. The reason
of this aspect ratio dependency will be discussed in Sec. 5.1.
300
(b)
250
200
m = 0.1
150
m = 0.05
m = 0.02
100
50
0
0
0.2
0.4
0.6
0.8
969
25
m = 0.2
(a)
Misorientation Δθ (°)
Nominal compressive stress, σ / MPa
Crystal Plasticity Analysis of Development of Intragranular Misorientations due to Kinking in HCP Single Crystals
m = 0.05
m = 0.1
15
m = 0.2
10
5
0
1
m = 0.02
20
0
Nominal compressive strain (%)
0.2
0.4
0.6
0.8
1
Normalized position in z-direction x3 / l
250
25
(a)
(b)
200
Misorientation Δθ (°)
Nominal compressive stress, σ / MPa
Fig. 9 Effect of strain rate sensitivity on the macroscopic stress-strain curves and on the distribution of ¦ª in the model with ªini = 3°.
(a) Variation in stress-strain curves. (b) Line profiles of ¦ª at ¾zz = ¹1%.
150
100
10 × 10 × 10 μm
10 × 10 × 20 μm
50
10 × 10 × 10 μm
20
10 × 10 × 20 μm
15
10 × 10 × 40 μm
10
5
10 × 10 × 40 μm
0
0
0.2
0.4
0.6
0.8
0
1
0
0.2
0.4
0.6
0.8
1
Normalized position in z-direction x3 / l
Nominal compressive strain (%)
Fig. 10 Effect of aspect ratio of specimen on stress-strain curves and on the distribution of ¦ª in the model with ªini = 3°. (a) Stress-strain
curves. (b) Line profiles of ¦ª at ¾zz = ¹1%.
4.5 Effect of boundary conditions
The deformed Zn single crystal in Fig. 1(a) shows that
kink bands are formed near the top and bottom regions of the
specimen, while the kink bands calculated in the present
analyses are formed in the center of the model. With the aim
of more realistically simulating the experimental observation
in Fig. 1, constraints on the displacement at the top or/and
bottom surfaces are introduced into the model subjected to
compression with ªini = 3° (TB-free model) as
near-½1120
shown in the previous sections.
Figure 11(a) shows schematic diagrams of three boundary
conditions; T-fixed, B-fixed and TB-fixed conditions. Constraints on the displacement in the direction perpendicular to
the loading direction are added to the TB-free model in
Fig. 11(a). In the T-fixed model, the displacements in the xand y-directions of all nodes at the top surface are set to zero.
In the B-fixed model, the displacements in the x- and ydirections of all nodes at the bottom surface are set to zero.
In the TB-fixed model, the displacements in the x- and ydirections of all nodes at the top and bottom surfaces are
set to zero. These additional constraints correspond to the
boundary conditions due to friction between the specimen
and the compressive jig.
Figure 11(b) shows the distributions of ¦ª at a nominal
compressive strain of ¾zz = ¹1%. The distributions change
significantly with a slight change in the boundary conditions.
The distributions for the T-fixed and B-fixed models are
similar when one of them is rotated 180° about the x-axis. In
both distributions, significant variations of crystal orientation
(a)
(b)
z
z
x
T-fixed
y
B-fixed
TB-fixed
x
y T-fixed
B-fixed
Δθ [°] : -3
TB-fixed
19
Fig. 11 Effect of boundary conditions on distribution of ¦ª. (a) Boundary
conditions for T-fixed, B-fixed and TB-fixed models. (b) Distributions of
¦ª at ¾zz = ¹1%.
are introduced near the fixed end. In the TB-fixed model, the
crystal orientation clearly varies near both ends. These results
indicate that the additional constraints on the top and/or
bottom surfaces lead to intragranular misorientations near the
top or/and bottom surfaces. Additionally, in the TB-fixed
model, there are regions with slightly negative values of ¦ª
near the top and bottom ends.
The numerical result of the TB-fixed model shows similar
to experimental result shown in Fig. 1(a) while the other
numerical results presented in this study are quite different
from the experimental result. One possible reason of the
similarity is friction on top and bottom surfaces in the real
experimental condition. Even in experiment, similar results to
TB-free model have been reported17,18) where the effect of
end constraints has been avoided by using longer specimens
compared with the one in Fig. 1(a).
970
T. Mayama, T. Ohashi, Y. Tadano and K. Hagihara
4.6 Compressive loading under double slip
In this section, calculations of compressive loading near
½0110
direction and near-intermediate direction between
and ½0110
directions are shown. In these loading
½1120
direction, two basal slip modes are activated.
The geometry of the model, the finite element mesh and
the boundary conditions are the same as those shown in
Fig. 2(a). Only the crystal orientation is changed in
accordance with the loading direction. The initial deviation
angle is set to be ªini = 3°. Figure 2(c) and (d) show
schematic illustrations of the crystal orientations. The
boundary condition for these models is TB-free.
Figure 12(a), (b) and (c) show the distributions of ¦ª after
compression of ¾zz = ¹1% nominal strain in near-½1120,
and near-intermediate directions, respectively.
near-½0110
compression in Fig. 12(b), the distribuFor the near-½0110
compression as shown
tion is similar to that for near-½1120
in Fig. 12(a) whereas the deformation modes are single slip
compression and double slip in the
in the case of near-½1120
compression. In contrast, as shown in
case of near-½0110
Fig. 12(c), the distribution of ¦ª for compression in near and ½0110
directions
intermediate direction between ½1120
compression and nearis clearly different from near-½1120
compression. The result of near-intermediate com½0110
pression shows orientation gradient along x-direction while
the orientation gradient along x-direction is negligible in
the results of near-½1120
compression and near-½0110
compression. The orientation gradient for near-intermediate
compression would be caused by different activities for nonequivalent basal slip modes.
5.
5.1
Discussion
Mechanism of development of intragranular misorientations
In the results of the calculations shown above, local crystal
orientations are significantly changed during compression
in single crystals. The use of the model without initial
fluctuations or imperfections is a crucial difference from
previous crystal plasticity analyses on kinking.8,10) When the
values of ªini, h and m are small, more significant intragranular misorientations develop. Figures 6 and 7 indicate
that the distributions of accumulation of basal slip and
intragranular misorientations are very similar. Here, the
mechanism of the development of intragranular misorientations is discussed in detail.
Figures 13(a)³(c) show the distributions of RSS for BSL2
at points A, B, and C indicated in Fig. 4. The ranges of the
contours are adjusted for each point. At point C, the RSS is
clearly concentrated in the center of the model. At point B,
the distribution of RSS is also nonuniform while the
concentration of RSS in the center of the model is weaker.
Even at point A, where the stress is increasing almost
linearly, the distribution of RSS is not uniform but the
difference between the maximum and minimum values of
RSS is small.
The nonuniform distributions of RSS in Figs. 13(a)³(c)
are caused by the nonuniform stress components. Only stress
components ·yy, ·yz and ·zz contribute to the RSS for BSL2
because the slip direction and slip plane normal direction for
(a)
(b)
y
Δθ [°] : 0
(c)
z
x
18
Fig. 12 Distribution of ¦ª at ¾zz = ¹1% in TB-free model compressed
direction, (b) near-½0110
in (a) near-½1120
direction and (c) near and ½0110.
intermediate direction between ½1120
(a)
(b)
(c)
9.4268
11.1
16.1
(d)
Lower
z
x
9.4262
[MPa]
10.7
[MPa]
Absolute
stress
value
9.1
[MPa]
Higher
σ zz
y
Fig. 13 Distributions of the resolved shear stress (RSS) on the basal slip
system BSL2 when the nominal compressive strain is at (a) A: 0.4%,
(b) B: 0.46% and (c) C: 0.48% in Fig. 4, respectively. (d) Schematic
diagram of distribution of stress component ·zz by enhancing the change
in the shape when the macroscopic stress is at point A in Fig. 4.
BSL2 lie on the y-z plane. It was confirmed that the stress
component ·zz shows the most significant heterogeneity in
the stress distribution at point A. Figure 13(d) shows a
schematic diagram of distribution of stress component ·zz by
enhancing the change in the shape of specimen. The reason
for this nonuniform distribution of stress component ·zz can
be explained in terms of the change in the shape of the
specimen. The slip direction and slip plane normal direction
of BSL2 in the model with ªini = 3° are s(BSL2)(0, ¹0.05234,
0.99863) and m(BSL)(0, ¹0.99863, ¹0.05234), respectively.
The Schmid tensor P(BSL2) for BSL2 is
1
PðBSL2Þ ¼ ðsðBSL2Þ mðBSL2Þ þ mðBSL2Þ sðBSL2Þ Þ
2
0
1
0
0
0
B
C
¼ @ 0 0:05226 0:49726 A:
ð3Þ
0 0:49726 0:05226
Because of the small deviation angle ªini between the loading
direction and the basal plane, the Schmid factor in the loading
direction (zz component of P(BSL2)) is nonzero, as indicated in
eq. (3). With compressive loading, the basal slip system is
activated, even at point A in Fig. 4, because the slip rate £_
calculated by eq. (1) is nonzero when the RSS is nonzero due
to the nonzero Schmid factor in the loading direction. Once
the basal slip system BSL2 is activated, the shape of the
specimen is changed by plastic strain. The plastic strain rate
Dp is calculated by
_ ðBSL2Þ :
Dp ¼ £P
ð4Þ
From £_ and eqs. (3) and (4), the yz component of shear strain
is induced. Owing to the yz component of shear strain, the
original square prism shape of the model is changed to an
oblique square prism shape as shown in Fig. 13(d), in which
broken lines show the original undeformed shape. The
oblique planes on both sides of the model, as shown in
Crystal Plasticity Analysis of Development of Intragranular Misorientations due to Kinking in HCP Single Crystals
Fig. 13(d), lead to nonuniform stress distributions in the
model to satisfy the balance of moments. It has been
confirmed by elastic plane stress finite element analysis that
similar distributions of stress components are induced in a
parallelogram subjected to uniform compressive displacement, corresponding to the two-dimensional model schematically shown in Fig. 13(d).
Next, the significance of lattice rotation in slip localization
is confirmed by the suppression of lattice rotation. In Fig. 4,
gray dashed-dotted line shows the calculated stress-strain
curve without lattice rotation. Without lattice rotation, stress
continues to increase.
The significant difference between the results with and
without lattice rotation is caused by geometrical softening.45)
In the present crystal plasticity model, the slip direction s(¡)
and slip plane normal direction m(¡) are updated by
_ ð¡Þ ¼ W mð¡Þ :
s_ð¡Þ ¼ W sð¡Þ and m
ð5Þ
In eq. (5), W* is the nonplastic (i.e., elastic and rigid body)
spin tensor given by
W ¼ W Wp ;
ð6Þ
where W and Wp are the continuum and plastic spin tensors,
respectively. When the plastic deformation Wp is caused by a
single slip of BSL2, the plastic spin tensor becomes
£_
Wp ¼ ðsðBSL2Þ mðBSL2Þ mðBSL2Þ sðBSL2Þ Þ
20
1
0
0
0
B
C
¼ £_ @ 0
0
0:5 A:
ð7Þ
0 0:5 0
When the plastic spin due to the BSL2 slip is sufficiently
large compared with the macroscopic spin W, W* is
approximately calculated by
W ¼ Wp :
ð8Þ
During compressive loading, £_ ðBSL2Þ becomes positive
according to eqs. (3) and (4). From eqs. (5), (7) and (8), it
is found that the deviation angle ª between the compressive
direction and the basal plane increases owing to the activation
of basal slip BSL2.
When the deviation angle ª is less than 45°, the lattice
rotation due to basal slip BSL2 increases the Schmid factor of
the basal slip system, which accelerates the activity of basal
slip BSL2. Even after large lattice rotation, the Schmid factor
of BSL2 is the largest among the three modes of the basal slip
system. Therefore, the dominant activity of BSL2 continues.
The effect of geometrical softening is especially pronounced in the case of smaller ªini. When ªini is smaller, the
Schmid factor in the loading direction becomes smaller. In
the present case, from eqs. (3) and (4), a larger slip rate £_ is
required in a model with a smaller Schmid factor in the zdirection (zz component of P(BSL2)) to induce a macroscopic
compressive plastic strain rate in the z-direction (zz
component of Dp). From eqs. (5), (7) and (8), a larger slip
rate leads to significant lattice rotation. The significant lattice
rotation results in the rapid increase in the Schmid factor
discussed in the previous paragraph. With the rapid increase
in the Schmid factor, the stress required to obtain the same
RSS decreases, which results in softening.
971
The dependence of the aspect ratio of model on intragranular misorientation as shown in Sec. 4.4 can be
explained by the above mentioned mechanism. When the
same amount of shear strain is induced for the models with
different aspect ratios, the displacement in y-direction on the
top surface as shown in Fig. 13(d) is dependent on the aspect
ratio. That is, the model with higher aspect ratio should show
larger displacement in y-direction on the top surface, which
lead to more significant nonuniform stress distributions as
shown in Fig. 13(d). As the result, the development of
intragranular misorientation is dependent on the aspect ratio
of the model as shown in Fig. 10.
From the similar consideration as discussed above, it is
found that lattice rotations in the opposite directions
cancelled out owing to the activation of two equivalent
compression in Sec. 4.6.
basal slip modes for near-½0110
Consequently, the axis of lattice rotation became the same in
compression and
the sample coordinates for both near-½1120
compression.
near-½0110
5.2
Suggested influences of present and potential
effective factors
Figures 4, 7, 8, 9 and 10 show variations of stress-strain
responses and the line profiles of developed intragranular
misorientations depending on the effective factors. One
common feature is that slight changes in parameters result in
significant change in the development of intragranular
misorientation. The changes are especially significant from
ªini = 4° to ªini = 5° as shown in Fig. 7(b), from h =
100 MPa to h = 200 MPa as shown in Fig. 8(b) and from
m = 0.05 to m = 0.1 as shown in Fig. 9(b). This fact
suggests that threshold values for each parameter to develop
significant misorientations could be exist.
The other interesting feature as shown in Figs. 4, 7, 8, 9 and
10 is the influence of effective factors on the shape of the line
profile of crystal orientation. The initial crystal orientation and
material parameters (hardening rate and strain rate sensitivity)
similarly affect the transitions of crystal orientation as shown
in line profiles even though the changes in stress-stress
response are qualitatively very different. Contrary, the effect
of aspect ratio on the transition of crystal orientation is
qualitatively different to the effect of other parameters.
Therefore, due to the superposition of effects of geometry
and the other parameters could lead to more variations of
development of intragranular misorientation as well as the
significant influences of boundary condition and activity of
each deformation mode as shown in the Sec. 4.5 and 4.6.
In the results of the present analysis, the transitions of
crystal orientations due to kinking are triangular which is
similar to the experimental result of “L2” in Fig. 1(h). As
pointed out in Sec. 2, the step-like transitions as “L1” in
Fig. 1(h) are frequently observed in the experiments. To
reproduce the step-like transitions, the numerical approaches
other than the present approach might be required. As
mentioned in Sec. 3.1, in the present study, non-local effects
are not taken into account. With the non-local effects, strain
hardening behaviour near kinked region should be very
different due to the development of non-local back stress
because there are pronounced slip gradient as shown in
Fig. 6, which affects slip distributions and intergranular
972
T. Mayama, T. Ohashi, Y. Tadano and K. Hagihara
misorientations. Relating to this, with the non-local effects,
the prediction of size dependence in kinking is also expected.
For further profound investigations with the aim of direct
comparisons to experimental results, therefore, non-local
effect should be included in the numerical method. Another
point to note is that the present crystal plasticity analysis is
performed based on the implicit assumption of the existence
of enough initial dislocations to induce heterogeneous
deformation field. This assumption is not adequate for the
kinking at smaller scale where the initial dislocation density
is not enough without nucleation of dislocations. In that case,
alternative or additional mechanisms should be considered
based on discrete approaches such as molecular dynamics
simulations.46)
6.
Conclusions
In this study, the development of intragranular misorientations in single crystals due to the dominant activity of the
basal slip system was investigated by a crystal plasticity finite
element method. The effects of the initial crystal orientation,
strain hardening rate, and strain rate sensitivity on the
development of intragranular misorientations were numerically evaluated. The mechanism of the development of
intragranular misorientations in models with no initial
fluctuations or imperfections was discussed in terms of the
nonuniform stress distribution and geometrical softening. The
effects of a slight change in the boundary conditions on
intragranular misorientations were also examined. The results
of the present numerical study are summarized as follows.
(1) Even in the case of no initial fluctuations or
imperfections, basal slip accumulated locally in a single
crystal. The localization originated from the nonuniform stress distribution caused by changes in the shape
from a square prism to an oblique square prism due to
the activation of the basal slip system. As a result of the
localized accumulation of basal slip, significant intragranular misorientations developed owing to the
continuous lattice rotation in the same direction.
(2) Without lattice rotation, basal slip accumulated much
more uniformly than in the case of localized accumulation with lattice rotation. The localized accumulation
of basal slip was accelerated by lattice rotation because
the lattice rotation due to basal slip increases the
Schmid factor for active basal slip systems.
(3) The initial deviation angle between the basal plane and
the compressive direction, the strain hardening rate, and
the rate sensitivity affected the development of intragranular misorientations. Smaller deviation angles, a
lower strain hardening rate and lower rate sensitivity
resulted in more significant development of intragranular misorientations.
Acknowledgments
This work was partially supported by a Grants-in-Aid
for Scientific Research from MEXT (No. 26109717 and
No. 26420020), JSPS Core-to-Core Program B, JSPS Program for Advancing Strategic International Networks to
Accelerate the Circulation of Talented Researchers, and the
Corporative Research Program of “Network Joint Research
Center for Materials and Devices”.
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