Simple shear and turbulence in the metals
The paper shows that metal in microvolumes behaves as turbulent flow during large plastic deformation under simple shear scheme. This gives a unified explanation of the following effects: saturation of strain hardening; abnormally fast diffusion in the metals under large plastic deformation; specime...
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| Zitieren: | Simple shear and turbulence in the metals / Y. Beygelzimer // Физика и техника высоких давлений. — 2010. — Т. 20, № 1. — С. 26-32. — Бібліогр.: 17 назв. — англ. |
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| citation_txt | Simple shear and turbulence in the metals / Y. Beygelzimer // Физика и техника высоких давлений. — 2010. — Т. 20, № 1. — С. 26-32. — Бібліогр.: 17 назв. — англ. |
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| description | The paper shows that metal in microvolumes behaves as turbulent flow during large plastic deformation under simple shear scheme. This gives a unified explanation of the following effects: saturation of strain hardening; abnormally fast diffusion in the metals under large plastic deformation; specimen lengthening in free end torsion test (known as the Swift effect); equalization of metal properties in different directions after sufficiently many passes of Twist Extrusion.
Изложена гипотеза о вихревом течении металлов при большой пластической деформации по схеме простого сдвига. На ее основе с единой точки зрения дана трактовка следующих эффектов: предела деформационного упрочнения в процессах интенсивной пластической деформации; так называемой аномально быстрой диффузии в пластически деформируемых металлах; удлинения образцов при кручении со свободными концами (Swift effect); выравнивания свойств металлов по различным направлениям при большом числе проходов методом винтовой экструзии. Приведены результаты экспериментов, свидетельствующие в пользу выдвинутой гипотезы.
Висунуто гіпотезу про вихрову течію металів при великій пластичній деформації за схемою простого зсуву. На її основі з єдиної точки зору подано трактування наступних ефектів: межі деформаційного зміцнення в процесах інтенсивної пластичної деформації; так званої аномально швидкої дифузії при пластичнiй деформації; подовження зразків при крученні з вільними кінцями (Swift effect); вирівнювання властивостей металів за різними напрямками при великій кількості проходів методом гвинтової екструзії. Наведено результати експериментів, що свідчать на користь висунутій гіпотезі.
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Физика и техника высоких давлений 2010, том 20, № 1
© Y. Beygelzimer, 2010
PACS: 62.20.Fe
Y. Beygelzimer
SIMPLE SHEAR AND TURBULENCE IN THE METALS
Donetsk Institute for Physics and Engineering named after A.A. Galkin of the NASU
72 R. Luxemburg St., Donetsk, 83114, Ukraine
Received November 25, 2009
The paper shows that metal in microvolumes behaves as turbulent flow during large
plastic deformation under simple shear scheme. This gives a unified explanation of the
following effects: saturation of strain hardening; abnormally fast diffusion in the metals
under large plastic deformation; specimen lengthening in free end torsion test (known as
the Swift effect); equalization of metal properties in different directions after sufficiently
many passes of Twist Extrusion.
Keywords: Simple shear, Severe plastic deformation, Swift effect, Turbulence, mixing
Introduction
Simple shear test are done by twisting cylindrical specimen and are widely
used for determining metal flow lines. There are two problems with transferring
the results of these experiments to other types of deformation. The first problem is
choosing a measure for equivalent strain that would let one combine the results of
shear and tension experiments into a single «unified flow curve». There is no une-
quivocal solution in the literature. A short but comprehensive survey of this sub-
ject appears in [1].
The second, more serious problem was revealed by Bridgeman [2] in his high
pressure experiments. He discovered that in large deformations, the stress-strain
curves for tension and shear diverge sharply: the first one continually increases
while the second one saturates. The same saturation effect in metal strengthening
during deformation according to simple shear scheme was also observed in High
Pressure Torsion (HPT) [3], Equal Channel Angular Pressing (ECAP) [4] and
Twist Extrusion (TE) [5].
In this work, we give an explanation of this effect based on a hypothesis that
metal in microvolumes behaves as turbulent flow [6,7]. This concept allows us to
give a unified explanation of the following effects related to deformation accord-
ing to simple shear scheme: saturation of strain hardening [2]; abnormally fast dif-
fusion in the metals under large plastic deformation [5,8]; specimen lengthening
Физика и техника высоких давлений 2010, том 20, № 1
27
in free end torsion test (known as the Swift effect) [9]; equalization of metal prop-
erties in different directions after sufficiently many passes of TE [10].
Note that we consider cold deformation, i.e., we assume that once loading stops
all processes in the metal stop as well.
1. External and internal vorticity
Simple shear is defined using the following velocity field V:
γxV y= , 0== zy VV , (1)
where γ is the shear strain rate, the axis x is the shear direction, the axis y is nor-
mal to the shear plane xz. Every material point of this field has a non-zero vortic-
ity with absolute value γ . A non-zero vorticity can be either due to the curvature
of the point’s trajectory (we called external vorticity), or due to a vortex motion
inside the material point (we called internal vorticity). In order to picture internal
vorticity, recall that each material point is the representative volume element
(RVE) of a polycrystal, which contains a great number of structural elements [11].
There are regions of polycrystal where dislocations get plugged during plastic de-
formation. Such regions cause bending of the crystalline lattice. Its relaxation im-
plies rotation, grains refinement and growth of the misorientation angle [12,13]. The
transformations above can be described using a random vortex velocity field v, ex-
isting inside a representative volume element. As a quantitative measure of this in-
ternal vorticity we take the average value of the velocity curl inside the RVE:
1curl curl d
S
S
S
= ⋅∫∫v v n .
Here angle brackets denote the average with respect to the RVE, integration is
done along the cross-section of the RVE, n is the unit normal vector of the cross-
section, and S is the area of the cross-section.
In [6,7], we establish an analogy between velocity field v and a velocity field
of a turbulent liquid flow. The two velocity fields have the same structure since
they are formed via gradual self-similar fragmentation of larger curls into in-
creasingly smaller curls. Furthermore, both fields have a limiting curl size when
fragmentation stops. In metals, this phenomenon is caused by grain boundary
sliding, while in liquids it is caused by viscous friction [14].
Under simple shear, according to Eq. (1), trajectories of material points are
rectilinear. This leads to two important conclusions: (a) under simple shear, vor-
ticity is only internal; (b) to realize simple shear, a material must be able to sup-
port an internal vorticity equal to shear strain rate. We argue that under internal
vorticity constraints, solid bodies cannot be deformed via simple shear. Internal
vortex is replaced with external vortex. This curves the trajectory of material
points. As a result, deformation becomes complex and non-homogenous. Thus we
will distinguish between «simple shear» and «deformation according to the simple
shear scheme».
Физика и техника высоких давлений 2010, том 20, № 1
28
We believe that the length changes during free end torsion test (Swift effect)
[9] is caused by annealed metals not being able to realize simple shear. As a re-
sult, twisting the material produces a velocity field with an axial component,
which changes the length of the specimen.
2. The main idea
We are making the following hypothesis. Under «deformation according to the
simple shear scheme», metals successively go through two stages: (a) Develop-
ment of Turbulence when there are constraints on the internal vorticity,
curl γ 2k=v , k < 1; (b) Fully Developed Turbulence when internal vorticity is
unconstrained, curl γ=v , k = 1.
Parameter k is a coefficient characterizing the degree of rotational freedom in
the RVE. The value of k is determined by defects enabling rotations of the crys-
talline lattice (e.g., disclinations or non-equilibrium grain boundaries). Before
grain refinement starts, there are no such defects and thus k = 0. Once grain re-
finement starts, such defects start to appear, and their number grows as strain in-
creases. When k reaches 1, grain refinement stop, turbulence reaches the second
stage, initiating simple shear.
In liquids, the fully developed turbulence stage is stationary [14]. By analogy,
the velocity field of simple shear Eq. (1) and the random vortex velocity field v
are independent of strain during this stage, where strain acts as an analog of time.
This means that under constant axial pressure P, metals do not harden under sim-
ple shear, since flow stress should also be independent of strain [6].
Strain rate hardening is possible, as in viscous liquid. Moreover, if there is no
strain rate hardening, the material loses its stability under simple shear, and the
deformation gets localized in a thin layer [15].
The hypothesis about the two stages of deformation naturally explain results of
classical experiments of Henri Tresca about hole punching [16]. The first stage
corresponds to the hardening phase, while the second stage – to the liquid-like
flow of metals without strain hardening. The fate of the two stages discovered by
Tresca about 150 years ago turned out to be very different. The first stage is scru-
pulously studied to this day, while the second stage was immediately overlooked
and was rediscovered only relatively recently, when studying severe plastic de-
formation (SPD) processes.
3. The two stages of deformation under the simple shear scheme
Strain hardening was investigated for different loading processes. Paul Ludvik
was the first to propose and experimentally justify a hypothesis about the equiva-
lence of hardening via pure and simple shear. This is how the notion of equivalent
strain came into existence. We give a different explanation for the experimental
results observed by Henri Tresca and Paul Ludvik. When deforming a metal
specimen according to the simple shear scheme, there are two consecutive stages:
Физика и техника высоких давлений 2010, том 20, № 1
29
(a) a complex, inhomogeneous strained state that is not simple shear, and (b) sim-
ple shear. Stage (a) exists due to constraints on internal vorticity. A given total
curl is composed of internal vorticity plus external vorticity, which curves the
trajectory of material points making deformation complex and non-uniform.
These ideas explain the emergence of macroswirls during the early stages of HPT,
an interesting effect recently observed in [17]. Deformation during stage (a) cre-
ates additional degrees of rotational freedom, causing a gradual transition from
pure to simple shear. This explanation is in agreement with the experiments in
[17], which show that as the number of HPT revolutions increases, macroswirls
gradually disappear and a uniform simple shear strain pattern settles throughout
the disk.
The appearance of this «gradually disappearing» pure shear is mistakenly taken
as experimental evidence that simple and pure shear affect metals in the same
way. In fact, they do not. Therefore, such notions as «single stress-strain curve»
and «the equivalent strain» make sense only when strain is relatively small. Since
the seeming equivalence of different loading schemes is explained by the gradu-
ally disappearing pure shear, a good candidate for the role of equivalent strain
during stage (a) are Eichinger’s relations based on nonlinear mechanics [1]. They
take into account the lengthening and rotations of material fibers. When the shear
strain is in the range 0 ≤ γ ≤ 2, we can substitute Eichinger’s relations for
ε γ 3= , obtained via integrating strain rate γ 3 over time. Since the obtained
results are very similar, this relation can also be used to compute the equivalent
strain in this interval. During simple shear (b), its strain γ is analogous to time in
the theory of turbulence in liquids.
4. Deformation-induced intermixing
Strong evidence in support of the analogy is the deformation-induced inter-
mixing of different phases and inclusions [5,8]. This effect is one of the most im-
portant manifestations of a turbulent flow and is explained by active mixing [14].
We hypothesize that it is turbulent motion, rather than a large increase in the dif-
fusion coefficient, that causes rapid mixing during large plastic deformation.
We propose a simple computer model for investigating turbulent mixing. Sup-
pose that we have a polycrystal with an inclusion. We will represent the poly-
crystal by a matrix of zeros, with the inclusion corresponding to the set of ones in
the matrix (Fig. 1,a). Turbulent velocity field is represented as a sequence of ran-
dom curls superimposed on this figure. Each curl rotates the region under it by a
random angle. The Fig. 1,b shows how the matrix looks after 100 steps. It’s easy
to see that such motion mixes the material quite rapidly.
5. Several implications of the proposed model related to SPD processes
Let’s analyze several implications of the proposed model related to SPD proc-
esses based on simple shear (HPT, ECAP, TE). It’s well known that strain hard-
ening has a limit in these processes. Here is why. Deformation according to the
Физика и техника высоких давлений 2010, том 20, № 1
30
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0
0 0 0 0 0 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0
0 0 0 0 0 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0
0 0 0 0 0 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0
0 0 0 0 0 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0
0 0 0 0 0 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0
0 0 0 0 0 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0
0 0 0 0 0 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0
0 0 0 0 0 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
a b
Fig. 1. A two-dimensional model for investigating turbulent mixing in the metals: a –
polycrystal with an inclusion (shown as a block of 1s) and a curl (the boundary of a curl
shown as dashed line with arrows); b – inclusion after 100 successive curls
simple shear scheme gradually creates a turbulent motion inside the RVE. Rota-
tional freedom increases and the material gradually transitions from the first stage
to the stage of Fully Developed Turbulence. The deformation gradually trans-
forms into simple shear, under which there is no strain hardening.
If the answer is correct then the ultrafine grained (UFG) materials obtained
with SPD processes should showcase two interesting effects.
Effect 1: Materials with a structure corresponding to the Fully Developed Tur-
bulence stage must be isotropic.
It has been convincingly shown that the velocity field of the Fully Developed
Turbulence stage must be locally isotropic [12]. If the proposed analogy is valid,
metals under SPD will eventually become isotropic (under very large strain). We
noted the isotropy tendencies on Al in [10]. Fig. 2 shows how the hardness of CP
titanium specimen in two orthogonal directions experimentally depends on num-
ber of TE passes. Fig. 2 shows the flattening of hardness in the two directions,
strongly supporting the hypothesis we made above.
Effect 2: The Swift effect should disappear in UFG materials
Indeed, by gaining a UFG structure via deformation according to the simple
shear scheme, metals transition into the stage of Fully Developed Turbulence.
Rotations are localized inside the material points, streamlines straighten during
free end torsion test, and the swift effect should disappear.
Fig. 3 shows free end torsion test results for Al alloy specimen after two passes
of twist extrusion. As we can see, the increase in gauge length LΔ is much
weaker here than in control, annealed specimen.
The arguments above lead to the following conclusion.
Conclusion
Due to constraints on internal vorticity, a complex stress-strain state is realized
in the specimen volume during early stages of deformation according to the simple
shear scheme; this is not simple shear. A given total curl is composed of internal
Физика и техника высоких давлений 2010, том 20, № 1
31
Fig. 2. A hardness of CP titanium specimen in two orthogonal directions depends on
number of TE passes: – initial state, ● – axis direction, ▲ – cross-section direction.
Twist extrusion temperature – 100°C
Fig. 3. Free end torsion test results for Al–3 wt.% Mg–0.3 wt.% Sc alloy: –□– – initial, –○– –
after TE. Twist extrusion temperature – 100°C. Gauge length – 38 mm, diameter – 5 mm
vorticity plus external vorticity, which curves the trajectory of material points
making deformation complex and non-uniform. Increasing strain when deforming
according to the simple shear scheme creates new degrees of rotational freedom.
This creates a flow in the material, a process similar to the development of turbu-
lence in liquids. When strain is sufficiently large, constraints on internal vorticity
disappear, leading to simple shear and the formation of a stationary UFG struc-
ture. Deformation hardening disappears, and the movement of metal in the RVE
becomes analogous to fully developed turbulence. UFG structures thus represent
«turbulence snapshots» of solid bodies. A turbulent motion can quickly transfer a
substance inside a solid body, much more efficiently than diffusion. This is of
both practical and theoretical interest.
The author thanks O. Prokof’eva, R. Kulagin, V. Bakhmatsky and M. Orlov for
the help in experiments.
1. S.C. Shrivastava, J.J. Jonas, G. Canova, J. Mech. Phys. Solids 30, No 1/2, 75 (1982).
2. P.W. Bridgman, Studies in Large Plastic Flow and Fracture with Special Emphasis on
the Effects of Hydrostatic Pressure, McGraw-Hill, New York–Toronto–London
(1952).
3. A.P. Zhilyaev, T.G. Langdon, Prog. Mater. Sci. 53, 893 (2008).
4. R.Z. Valiev, T.G. Langdon, Prog. Mater. Sci. 51, 881 (2006).
5. Y. Beygelzimer, V. Varyukhin, S. Synkov, D. Orlov, Mater. Sci. Eng. A503, 14 (2009).
6. Y. Beygelzimer, Fiz. Tekh. Vys. Davl. 18, No 4, 77 (2008) [in Russian].
7. Y. Beygelzimer, Mechanics of Materials 37, 753 (2005).
8. X. Sauvage, R. Pippan, Mater. Sci. Eng. A410–411, 345 (2005).
9. L. Ducheˆne, F.E. Houdaigui, A.M. Habraken, Int. J. Plasticity 23, 1417 (2007).
Физика и техника высоких давлений 2010, том 20, № 1
32
10. D. Orlov, Y. Beygelzimer, S. Synkov, V. Varyukhin, N. Tsuji and Z. Horita, Mater.
Sci. Eng. A519, 105 (2009).
11. S. Nemat-Nasser, M. Hori, Micromechanics: Overall Properties of Heterogeneous
Materials, Elsevier, Amsterdam (1999).
12. V.V. Rybin, High plastic deformation and fracture of metals, Metallurgia, Moscow
(1986) [in Russian].
13. N. Hansen and D. Jensen, Phil. Trans. R. Soc. Lond. A357, 1447 (1999).
14. A.S. Monin, A.M. Yaglom, Mechanics of Turbulence, Vol. 1, Dover Publications
(2007).
15. Y. Beygelzimer, B. Efros, V. Varyukhin, A. Khokhlov, Engng. Fract. Mech. 48, 629
(1994).
16. D.F. Bell, The Encyclopedia of Physics VI a (1). In: Mechanics of Solids 1, C. Trus-
dell (Ed.), Springer-Verlag, Berlin (1973).
17. Y. Cao, Y.B. Wang, S.N. Alhajeri, X.Z. Liao, W.L. Zheng, S.P. Ringer, T.G. Langdon
and Y.T. Zhu, J. Mater. Sci., DOI:10.1007/s10853-009-3998-2
Я.Ю. Бейгельзимер
ПРОСТИЙ ЗСУВ ТА ТУРБУЛЕНТНІСТЬ В МЕТАЛАХ
Висунуто гіпотезу про вихрову течію металів при великій пластичній деформації за
схемою простого зсуву. На її основі з єдиної точки зору подано трактування на-
ступних ефектів: межі деформаційного зміцнення в процесах інтенсивної пластич-
ної деформації; так званої аномально швидкої дифузії при пластичнiй деформації;
подовження зразків при крученні з вільними кінцями (Swift effect); вирівнювання
властивостей металів за різними напрямками при великій кількості проходів мето-
дом гвинтової екструзії. Наведено результати експериментів, що свідчать на ко-
ристь висунутій гіпотезі.
Ключові слова: простий зсув, інтенсивна пластична деформація, свіфт-ефект, тур-
булентність, перемішування
Я.Е. Бейгельзимер
ПРОСТОЙ СДВИГ И ТУРБУЛЕНТНОСТЬ В МЕТАЛЛАХ
Изложена гипотеза о вихревом течении металлов при большой пластической де-
формации по схеме простого сдвига. На ее основе с единой точки зрения дана трак-
товка следующих эффектов: предела деформационного упрочнения в процессах
интенсивной пластической деформации; так называемой аномально быстрой диф-
фузии в пластически деформируемых металлах; удлинения образцов при кручении
со свободными концами (Swift effect); выравнивания свойств металлов по различ-
ным направлениям при большом числе проходов методом винтовой экструзии.
Приведены результаты экспериментов, свидетельствующие в пользу выдвинутой
гипотезы.
Ключевые слова: простой сдвиг, интенсивная пластическая деформация, свифт-
эффект, турбулентность, перемешивание
|
| id | nasplib_isofts_kiev_ua-123456789-69259 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 0868-5924 |
| language | English |
| last_indexed | 2025-12-07T15:35:56Z |
| publishDate | 2010 |
| publisher | Донецький фізико-технічний інститут ім. О.О. Галкіна НАН України |
| record_format | dspace |
| spelling | Beygelzimer, Y. 2014-10-09T19:12:41Z 2014-10-09T19:12:41Z 2010 Simple shear and turbulence in the metals / Y. Beygelzimer // Физика и техника высоких давлений. — 2010. — Т. 20, № 1. — С. 26-32. — Бібліогр.: 17 назв. — англ. 0868-5924 PACS: 62.20.Fe https://nasplib.isofts.kiev.ua/handle/123456789/69259 The paper shows that metal in microvolumes behaves as turbulent flow during large plastic deformation under simple shear scheme. This gives a unified explanation of the following effects: saturation of strain hardening; abnormally fast diffusion in the metals under large plastic deformation; specimen lengthening in free end torsion test (known as the Swift effect); equalization of metal properties in different directions after sufficiently many passes of Twist Extrusion. Изложена гипотеза о вихревом течении металлов при большой пластической деформации по схеме простого сдвига. На ее основе с единой точки зрения дана трактовка следующих эффектов: предела деформационного упрочнения в процессах интенсивной пластической деформации; так называемой аномально быстрой диффузии в пластически деформируемых металлах; удлинения образцов при кручении со свободными концами (Swift effect); выравнивания свойств металлов по различным направлениям при большом числе проходов методом винтовой экструзии. Приведены результаты экспериментов, свидетельствующие в пользу выдвинутой гипотезы. Висунуто гіпотезу про вихрову течію металів при великій пластичній деформації за схемою простого зсуву. На її основі з єдиної точки зору подано трактування наступних ефектів: межі деформаційного зміцнення в процесах інтенсивної пластичної деформації; так званої аномально швидкої дифузії при пластичнiй деформації; подовження зразків при крученні з вільними кінцями (Swift effect); вирівнювання властивостей металів за різними напрямками при великій кількості проходів методом гвинтової екструзії. Наведено результати експериментів, що свідчать на користь висунутій гіпотезі. The author thanks O. Prokof’eva, R. Kulagin, V. Bakhmatsky and M. Orlov for the help in experiments. en Донецький фізико-технічний інститут ім. О.О. Галкіна НАН України Физика и техника высоких давлений Simple shear and turbulence in the metals Простой сдвиг и турбулентность в металлах Простий зсув та турбулентність в металах Article published earlier |
| spellingShingle | Simple shear and turbulence in the metals Beygelzimer, Y. |
| title | Simple shear and turbulence in the metals |
| title_alt | Простой сдвиг и турбулентность в металлах Простий зсув та турбулентність в металах |
| title_full | Simple shear and turbulence in the metals |
| title_fullStr | Simple shear and turbulence in the metals |
| title_full_unstemmed | Simple shear and turbulence in the metals |
| title_short | Simple shear and turbulence in the metals |
| title_sort | simple shear and turbulence in the metals |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/69259 |
| work_keys_str_mv | AT beygelzimery simpleshearandturbulenceinthemetals AT beygelzimery prostoisdvigiturbulentnostʹvmetallah AT beygelzimery prostiizsuvtaturbulentnístʹvmetalah |