Modeling of grain subdivision during severe plastic deformation by VPSC method combined with disclination analysis
Microstructure development during severe plastic deformation by simple shear is modeled using a combination of the visco-plastic self-consistent (VPSC) method and a disclination model. Strain incompatibilities between a homogeneous effective medium and a grain are calculated by VPSC. These are assum...
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| Zitieren: | Modeling of grain subdivision during severe plastic deformation by VPSC method combined with disclination analysis / A.A. Nazarov, N.A. Enikeev, A.E. Romanov, T.S. Orlova, I.V. Alexandrov, I.J. Beyerlein // Физика и техника высоких давлений. — 2005. — Т. 15, № 1. — С. 31-37. — Бібліогр.: 15 назв. — англ. |
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| author | Nazarov, A.A. Enikeev, N.A. Romanov, A.E. Orlova, T.S. Alexandrov, I.V. Beyerlei, n I.J. |
| author_facet | Nazarov, A.A. Enikeev, N.A. Romanov, A.E. Orlova, T.S. Alexandrov, I.V. Beyerlei, n I.J. |
| citation_txt | Modeling of grain subdivision during severe plastic deformation by VPSC method combined with disclination analysis / A.A. Nazarov, N.A. Enikeev, A.E. Romanov, T.S. Orlova, I.V. Alexandrov, I.J. Beyerlein // Физика и техника высоких давлений. — 2005. — Т. 15, № 1. — С. 31-37. — Бібліогр.: 15 назв. — англ. |
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| container_title | Физика и техника высоких давлений |
| description | Microstructure development during severe plastic deformation by simple shear is modeled using a combination of the visco-plastic self-consistent (VPSC) method and a disclination model. Strain incompatibilities between a homogeneous effective medium and a grain are calculated by VPSC. These are assumed to result in an accumulation of disclinations in the junctions of a grain that are relaxed by a growth of low-angle dislocation boundaries from the junctions. Predicted misorientation distributions between subgrains and their parent grains agree semi-quantitatively with experimental misorientation distributions for geometrically necessary boundaries. The texture after 100% simple shear was found to be insensitive to the presence of subgrains with misorientations less than 15º.
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Физика и техника высоких давлений 2005, том 15, № 1
31
PACS: 61.63.Bn
A.A. Nazarov1,2, N.A. Enikeev1, A.E. Romanov3, T.S. Orlova3,
I.V. Alexandrov1, I.J. Beyerlein4
MODELING OF GRAIN SUBDIVISION DURING SEVERE PLASTIC
DEFORMATION BY VPSC METHOD COMBINED WITH
DISCLINATION ANALYSIS
1Ufa State Aviation Technical University
Ufa, 450000, Russia
2Institute for Metals Superplasticity Problems, Russian Academy of Sciences
Ufa, 450001, Russia
3Ioffe Physico-Technical Institute, Russian Academy of Sciences
St. Petersburg, 194021, Russia
4Los Alamos National Laboratory
Los Alamos, NM 87545, USA
Microstructure development during severe plastic deformation by simple shear is mod-
eled using a combination of the visco-plastic self-consistent (VPSC) method and a discli-
nation model. Strain incompatibilities between a homogeneous effective medium and a
grain are calculated by VPSC. These are assumed to result in an accumulation of discli-
nations in the junctions of a grain that are relaxed by a growth of low-angle dislocation
boundaries from the junctions. Predicted misorientation distributions between subgrains
and their parent grains agree semi-quantitatively with experimental misorientation dis-
tributions for geometrically necessary boundaries. The texture after 100% simple shear
was found to be insensitive to the presence of subgrains with misorientations less than 15º.
Introduction
Grain fragmentation and refinement is one of the most important consequences
of dislocation substructure evolution during large plastic deformation common for
crystals of different types [1,2]. Mechanisms and characteristics of microstructural
refinement is a matter of great theoretical and practical importance, particularly in
producing ultrafine grained materials using equal-channel angular pressing
(ECAP) and other methods of severe plastic deformation [3]. In principle, the re-
finement process can be rigorously described in terms of dislocation dynamics,
though full 3-D modeling is a challenging task due to the complex nature of dislo-
cation multiplication, interaction and annihilation in real crystals. Recently, the
Физика и техника высоких давлений 2005, том 15, № 1
32
disclination approach has been shown to be very helpful for the description of
elementary processes of grain subdivision [1,4]. It has been proposed that grains
subdivide by the growth of low-angle boundaries due to the motion of partial
disclinations generated at grain boundaries and grain boundary junctions. In turn,
this formalism alone cannot be used to analyze the whole process of grain subdi-
vision, since it neglects the distribution of plastic deformation on slip systems and
does not consider important crystallographic and geometric factors influencing
grain refinement.
On the other hand, polycrystal models are successfully used to study the plastic
deformation of polycrystals and texture formation. These models account for the
distribution of strain among crystallographic slip systems and grains, albeit as an
approximate continuum approach. The VPSC model is one such approach [5].
This model considers each grain as an ellipsoidal inclusion deforming in a homo-
geneous equivalent medium (HEM) representing the whole polycrystal. Very re-
cently, this model has been applied to simulate the texture evolution during ECAP
[6]. However, the one-site VPSC model used in the cited paper neglects the non-
uniformities of the stress and strain within grains that lead to subdivison. Instead, a
very simple geometric criterion of grain subdivision was used and thus internal
grain microstructural development was neglected.
The joint use of disclination models with polycrystal modeling combines the
advantages of both methods and makes a powerful tool for modeling microstruc-
ture development during large plastic deformation. In this combination, VPSC
provides the data on the incompatibilities of strain between the HEM and a grain,
while the disclination model predicts subgrain formation and characteristics. The
aim of the present paper is to develop such a combined simulation scheme for se-
vere plastic deformation. By modeling simple shear we will demonstrate that this
scheme allows one to describe the main microstructural characteristics of grain
subdivision, such as the evolution of misorientations and texture.
Description of the model
Following the basic ideas of the VPSC method, consider an ellipsoidal grain,
Fig. 1,b, deforming in a HEM, representing the average behavior of the aggregate.
Let the strain rate of the HEM be given by a tensor E& . For the case of simple
shear, Fig. 1,a, with a velocity gradient tensor in the 1, 2, 3 coordinate system
=
000
000
00 D
D , (1)
the strain rate of the HEM is 2/2112 DEE == && , equal to the applied strain rate.
The strain rate tensor of a grain is denoted as ε& and depends on its orientation. If
the grain deforms differently than the HEM, strain incompatibilities will result in
an accumulation of disclinations in the junctions of this grain that will force its
Физика и техника высоких давлений 2005, том 15, № 1
33
splitting. Splitting occurs by redistribution of slip in different domains of the grain
and formation of low-angle dislocation boundaries. To be able to introduce junc-
tion disclinations and obtain a criterion for the splitting, we approximate the grain
by a parallelepiped circumscribing the corresponding ellipsoid (Fig. 1,b). Edges of
this parallelepiped coincide with the axes of ellipsoid. The difference between the
shear strain rates of the HEM and a grain, 1212 ε− &&E , are then transformed to the
coordinate frame made by the axes of the ellipsoid (1′, 2′, 3′) (Fig. 1,b). Let ijς
denote the components of this transformed tensor. Using the formulae in [7] one
can calculate the rates of disclination accumulation on the edges of the parallele-
piped as follows:
].2 ,,[],,2 ,[],,,2[ 121323312132321213231 ςςς=Ωςςς=Ωςςς=Ω &&& (2)
If 01212 >ε− &&E and when the maximum of Ω1, Ω2, Ω3 exceeds the critical
value of Ωc = 1°, it is assumed that a pair of dislocation boundaries with a misori-
entation vector θ = Ω1 + Ω2 + Ω3 is formed along the diagonals of the parallelepi-
ped as shown in Fig. 1,b. Full relaxation of junction disclinations and a rotation of
two of the four subgrains formed with this misorientation vector result (Fig. 1,c).
The sense of this rotation is chosen such that the subgrains rotate in a direction
nearly opposite to that of the parent grain. Nucleation of subgrain boundaries as
described above will be referred to as the splitting of a grain.
a b c
Fig. 1. A model of grain subdivision by a growth of a pair of subboundaries along the
grain diagonals: a − coordinate frame: 1 − shear direction, 2 − normal direction, 3 −
transverse direction; b − a junction disclination quadrupole and a mode of its relaxation; c −
new grains (subgrains) formed due to subdivision
Further straining accumulates new disclinations on the junctions unless the
strength of one of them exceeds Ωc. These subsequent disclinations will relax by
evolving along the initially formed low-angle boundaries causing a step-by-step
increase in their misorientation angle. The rotation matrix for these stepwise sub-
grain rotations is accumulated unless the angle of rotation becomes larger than a
value delimiting low- and high-angle grain boundaries (θ0 = 10° to 15°). When
Физика и техника высоких давлений 2005, том 15, № 1
34
this occurs, the grain is considered to subdivide into four grains, which are as-
signed corresponding orientations and further included into the aggregate simu-
lated by VPSC. In turn, these new grains can split and subsequently subdivide
during further straining thus leading to the formation of next-generation subgrains
and new grains. Thus, the number of grains will progressively increase during de-
formation. Each time a grain can contain only one generation of subgrains; new
generations will form when subgrains become independent grains and accumulate
their own internal misorientation.
Results of simulations
To test the proposed simulation scheme simple shear of a polycrystalline ag-
gregate containing initially 200 randomly oriented grains is modeled (Fig. 1,a).
The evolution of the microstructure is described in terms of the calculated number
of splitting/subdivison events per original grain, total number of split/sub-
divisions, misorientation distributions and texture for different values of applied
strain. The low/high angle boundary limit is assumed to be 15º.
For the aggregate a simulated shear strain of E12 = 1.0 (equivalent von Mises
strain EVM = 1.12) resulted in 30 subdivisions; one grain subdivided 6 times, some
grains 2 and 3 times, and others only once. The lower the strain rate of a grain, the
more rapidly it divides into subgrains. Accumulation of a 15º misorientation suffi-
cient for the first subdivision occurred after E12 = 0.36. Among 230 grains in the
final aggregate, 152 grains are split into subgrains with misorientations less than 15º.
For each new grain and subgrain, the identity of their parent grain is recorded,
which allowed a conditional misorientation distribution function f(θ) to be calcu-
lated. In calculating this function, misorientations θ of all new grains and sub-
grains with respect to the current orientation of their parent grain is taken into ac-
count. This function is calculated instead of the true distribution function, since in
the 1-site VPSC neighborship of grains is not defined. Nevertheless, f(θ) can give
semi-quantitative information on the distribution of misorientations of subbounda-
ries. Histograms representing the distribution for three values of the shear strain
are plotted in Fig. 2,a−c. In accordance with trends observed in experimental data
[8−10], the distribution becomes wider and the average misorientation increases
with increasing strain. The dependence of the average misorientation angle on the
strain plotted in Fig. 2,d agrees quantitatively well with experimental data ob-
tained on the substructure of cold-rolled metals [8−10].
Presented in Fig. 3 are the pole figures calculated for the polycrystal subjected
to E12 = 1.0. Fig. 3,a has been calculated using output from the standard VPSC
code, Fig. 3,b for the aggregate of 230 grains resulting from combined
VPSC/disclination simulations, i.e. taking into account the orientations of only
grains including those newly formed during deformation, while Fig. 3,c addition-
ally includes all subgrains rotated to less than 15º with respect to their parent
grains. Generally, during the simulation with grain subdivision texture develops in
the same way as in the standard VPSC code. Maxima are located along the same
Физика и техника высоких давлений 2005, том 15, № 1
35
a b
c d
Fig. 2. Evolution of the conditional misorientation distribution function during simple
shear strain up to EVM = 0.23 (а), 0.68 (b), and 1.12 (c) and a plot of the average misori-
entation angle of subgrains with strain (d)
a b c
Fig. 3. Pole figures for a polycrystal with 200 initially random grains deformed by simple
shear up to EVM = 1.12: а − after a standard VPSC simulation (number of grains 200); b −
after a VPSC/disclination simulation neglecting subgrains with misorientations θ < θ0 = 15°
(final number of grains 230); c − as in b, additionally including subgrains (final number
of grains and subgrains 382). Intensity level increment is equal to 0.95. Direction of
shearing and axes are defined in Fig. 1,a
Физика и техника высоких давлений 2005, том 15, № 1
36
directions and have nearly the same value for all three cases (9.54, 9.64, and 9.56,
respectively). It would be expected that the presence of low-angle misorientations
of up to 15º inside grains would result in a significant spread of the texture. Sur-
prisingly, however, only slight differences in the pole density in the vicinity of
some maxima are observed in Fig. 3,c as compared to Fig. 3,a. Nevertheless, all
the simulated pole figures are in a good qualitative agreement with well-known
experimental data on standard textures for simple shear (see, for example [15]).
This result may not be too surprising as standard VPSC has proven to predict well
first ECAP pass textures in many materials (See [13] and [14] and references therein).
However, the coupling between misorientation evolution and texture is expected to
become stronger under larger strains and strain path changes, such as in ECAP.
Discussion
We have presented a simple model, which combines the advantages of both the
VPSC polycrystal model and a disclination model for subgrain boundary nuclea-
tion and evolution and provides a possibility to study the substructure evolution
during large plastic deformation. The key feature of this model is a split-
ting/subdivision criterion that is based on the disclination concept.
The relaxation of junction disclinations occurs by a complicated process of re-
distribution of slip near the junctions under the combined action of applied and
internal stresses. This process has to be analyzed with a model that accounts for
the non-uniformity of stress and strain inside grains. This can be done in terms of
the n-site VPSC model [11] or a finite-element simulation. However for mul-
tiscale modeling, these methods would be computationally too expensive. Instead,
physically-based models can be used to understand the modes of relaxation of
disclinations. The model presented here is the simplest one of these. More sophis-
ticated models can be put forward on the basis of an energetic analysis of discli-
nation systems. The development of such models is under way [12]. Nevertheless,
even with the simplest disclination subdivision criterion, this combined approach
is capable of describing some important structural characteristics of heavily de-
formed polycrystals and thus can be used to simulate the ECAP process.
Recent electron microscope studies of cold-rolled polycrystals provided exten-
sive data on the substructure evolution during large deformation. One of the im-
portant findings of those works is that geometrically-necessary boundaries
(GNBs) form on two different scales [9]. The boundaries formed on a smaller
scale slightly larger than the cell size have misorientations less than 6°, whereas
the GNBs formed on a larger scale slightly smaller than the grain size can have
misorientations exceeding 15°. It is the latter type whose average misorientation
angle increases with strain in good accordance with the predictions presented in
Fig. 2,d [9]. One can suggest that the first type of GNBs forms due to an inhomo-
geneity of strain inside grains, while the second type forms due to the incompati-
bility of strain of neighboring grains, by an accumulation and relaxation of junc-
tion disclinations.
Физика и техника высоких давлений 2005, том 15, № 1
37
Thus, the model presented in this paper describes well the formation and pa-
rameters of the largest-scale substructure in deformed metals, which is character-
ized by truly large-angle misorientations and most important for grain refinement
by severe plastic deformation.
Finally, only slight differences of the final texture are found between the stan-
dard and modified VPSC codes for the case of simple shear up to 100%. How-
ever, for deformation with strain path changes, such as in multi-pass ECAP, a
more significant influence of grain subdivision on texture can be expected.
This research is funded by a Los Alamos National Laboratory – Initiatives for
Proliferation Prevention Project № T2-0197.
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7. V.V. Rybin, A.A. Zisman, N.Yu. Zolotarevsky, Acta Metall. Mater. 41, 2211 (1983).
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10. D.A. Hughes, N. Hansen, Acta Mater. 48, 2985 (2000).
11. R.A. Lebensohn, Acta Mater. 49, 2723 (2001).
12. T.S. Orlova, A.A. Nazarov, N.A. Enikeev, I.V. Alexandrov, R.Z. Valiev,
A.E. Romanov, Phys. Solid State, submitted.
13. S. Li, I.J. Beyerlein, C.T. Necker, D.J. Alexander, N.A.M. Bourke, Acta Mater. 52,
4859 (2004).
14. S.C. Vogel, D.J. Alexander, I.J. Beyerlein, M.A.M. Bourke, D.W. Brown, B. Clausen,
C.N. Tomé, R.B. Von Dreele, C. Xu, T.G. Langdon, Mater. Sci. Forum 426−432, 2661
(2003).
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Press, Cambridge (2000).
|
| id | nasplib_isofts_kiev_ua-123456789-70106 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 0868-5924 |
| language | English |
| last_indexed | 2025-12-07T16:11:09Z |
| publishDate | 2005 |
| publisher | Донецький фізико-технічний інститут ім. О.О. Галкіна НАН України |
| record_format | dspace |
| spelling | Nazarov, A.A. Enikeev, N.A. Romanov, A.E. Orlova, T.S. Alexandrov, I.V. Beyerlei, n I.J. 2014-10-28T19:24:30Z 2014-10-28T19:24:30Z 2005 Modeling of grain subdivision during severe plastic deformation by VPSC method combined with disclination analysis / A.A. Nazarov, N.A. Enikeev, A.E. Romanov, T.S. Orlova, I.V. Alexandrov, I.J. Beyerlein // Физика и техника высоких давлений. — 2005. — Т. 15, № 1. — С. 31-37. — Бібліогр.: 15 назв. — англ. 0868-5924 PACS: 61.63.Bn https://nasplib.isofts.kiev.ua/handle/123456789/70106 Microstructure development during severe plastic deformation by simple shear is modeled using a combination of the visco-plastic self-consistent (VPSC) method and a disclination model. Strain incompatibilities between a homogeneous effective medium and a grain are calculated by VPSC. These are assumed to result in an accumulation of disclinations in the junctions of a grain that are relaxed by a growth of low-angle dislocation boundaries from the junctions. Predicted misorientation distributions between subgrains and their parent grains agree semi-quantitatively with experimental misorientation distributions for geometrically necessary boundaries. The texture after 100% simple shear was found to be insensitive to the presence of subgrains with misorientations less than 15º. This research is funded by a Los Alamos National Laboratory – Initiatives for Proliferation Prevention Project № T2-0197. en Донецький фізико-технічний інститут ім. О.О. Галкіна НАН України Физика и техника высоких давлений Modeling of grain subdivision during severe plastic deformation by VPSC method combined with disclination analysis Article published earlier |
| spellingShingle | Modeling of grain subdivision during severe plastic deformation by VPSC method combined with disclination analysis Nazarov, A.A. Enikeev, N.A. Romanov, A.E. Orlova, T.S. Alexandrov, I.V. Beyerlei, n I.J. |
| title | Modeling of grain subdivision during severe plastic deformation by VPSC method combined with disclination analysis |
| title_full | Modeling of grain subdivision during severe plastic deformation by VPSC method combined with disclination analysis |
| title_fullStr | Modeling of grain subdivision during severe plastic deformation by VPSC method combined with disclination analysis |
| title_full_unstemmed | Modeling of grain subdivision during severe plastic deformation by VPSC method combined with disclination analysis |
| title_short | Modeling of grain subdivision during severe plastic deformation by VPSC method combined with disclination analysis |
| title_sort | modeling of grain subdivision during severe plastic deformation by vpsc method combined with disclination analysis |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/70106 |
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