Equivalent strain in simple shear of metals
We show that simple shear and pure shear form two groups of transformations with different properties. The equivalent strain is viewed as an external control parameter of the deformation process at low homologous temperatures. The von Mises strain satisfies group-theoretic properties of both groups,...
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Донецький фізико-технічний інститут ім. О.О. Галкіна НАН України
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irk-123456789-1073882016-10-20T03:02:19Z Equivalent strain in simple shear of metals Beygelzimer, Y. We show that simple shear and pure shear form two groups of transformations with different properties. The equivalent strain is viewed as an external control parameter of the deformation process at low homologous temperatures. The von Mises strain satisfies group-theoretic properties of both groups, supporting its use for measuring the equivalent strain. On the other hand, the Hencky strain does not satisfy the simple shear group properties, implying that it is not appropriate for measuring the equivalent strain in simple shear. The paper also proposes a hypothesis explaining the absence of metal hardening in large simple shear deformations. This hypothesis explains why excluding rotations as prescribed by the finite strain theory is not valid. Показано, що простий і чистий зсуви утворюють дві групи перетворень з різними властивостями. Розглянуто еквівалентну деформацію як керуючий параметр процесу деформування. Показано, що деформація Von Mises задовольняє груповим властивостями і простого, і чистого зсуву, в той час як деформація Hencky не задовольняє груповим властивостям простого зсуву. На основі цього зроблено висновок про непридатність деформації Hencky в якості еквівалентної деформації простого зсуву. Висловлено й обґрунтовано гіпотезу про причину відсутності зміцнення металів при простому зсуві великої величини, яка пояснює, чому коректне (у рамках механіки кінцевих деформацій) виключення повороту, що відповідний кінцевому зсуву, призводить до невірного результату. Показано, что простой и чистый сдвиги образуют две группы преобразований с разными свойствами. Рассмотрена эквивалентная деформация как управляющий параметр процесса деформирования. Показано, что деформация Von Mises удовлетворяет групповым свойствам и простого, и чистого сдвигов, в то время как деформация Hencky не удовлетворяет групповым свойствам простого сдвига. На основе этого сделан вывод о непригодности деформации Hencky в качестве меры эквивалентной деформации простого сдвига. Высказана и обоснована гипотеза о причине отсутствия упрочнения металлов при простом сдвиге большой величины, которая объясняет, почему корректное (в рамках механики конечных деформаций) исключение поворота, соответствующего конечному сдвигу, приводит к неверному результату. 2015 Article Equivalent strain in simple shear of metals / Y. Beygelzimer // Физика и техника высоких давлений. — 2015. — Т. 25, № 1-2. — С. 90-97. — Бібліогр.: 28 назв. — англ. 0868-5924 PACS: 62.20.Fe http://dspace.nbuv.gov.ua/handle/123456789/107388 en Физика и техника высоких давлений Донецький фізико-технічний інститут ім. О.О. Галкіна НАН України |
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We show that simple shear and pure shear form two groups of transformations with different properties. The equivalent strain is viewed as an external control parameter of the deformation process at low homologous temperatures. The von Mises strain satisfies group-theoretic properties of both groups, supporting its use for measuring the equivalent strain. On the other hand, the Hencky strain does not satisfy the simple shear group properties, implying that it is not appropriate for measuring the equivalent strain in simple shear. The paper also proposes a hypothesis explaining the absence of metal hardening in large simple shear deformations. This hypothesis explains why excluding rotations as prescribed by the finite strain theory is not valid. |
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Beygelzimer, Y. |
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Beygelzimer, Y. Equivalent strain in simple shear of metals Физика и техника высоких давлений |
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Beygelzimer, Y. |
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Beygelzimer, Y. |
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Equivalent strain in simple shear of metals |
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Equivalent strain in simple shear of metals |
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Equivalent strain in simple shear of metals |
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Equivalent strain in simple shear of metals |
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Equivalent strain in simple shear of metals |
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equivalent strain in simple shear of metals |
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Донецький фізико-технічний інститут ім. О.О. Галкіна НАН України |
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| citation_txt |
Equivalent strain in simple shear of metals / Y. Beygelzimer // Физика и техника высоких давлений. — 2015. — Т. 25, № 1-2. — С. 90-97. — Бібліогр.: 28 назв. — англ. |
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Физика и техника высоких давлений |
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2025-07-07T19:53:34Z |
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| fulltext |
Физика и техника высоких давлений 2015, том 25, № 1–2
© Я. Бейгельзимер, 2015
PACS: 62.20.Fe
Yan Beygelzimer
EQUIVALENT STRAIN IN SIMPLE SHEAR OF METALS
Received January 22, 2015
We show that simple shear and pure shear form two groups of transformations with
different properties. The equivalent strain is viewed as an external control pa-
rameter of the deformation process at low homologous temperatures. The von Mises
strain satisfies group-theoretic properties of both groups, supporting its use for
measuring the equivalent strain. On the other hand, the Hencky strain does not sat-
isfy the simple shear group properties, implying that it is not appropriate for meas-
uring the equivalent strain in simple shear. The paper also proposes a hypothesis
explaining the absence of metal hardening in large simple shear deformations. This
hypothesis explains why excluding rotations as prescribed by the finite strain theory
is not valid.
Keywords: simple shear, severe plastic deformation, equivalent strain, ultrafine grained
materials
Показано, що простий і чистий зсуви утворюють дві групи перетворень з різними
властивостями. Розглянуто еквівалентну деформацію як керуючий параметр про-
цесу деформування. Показано, що деформація Von Mises задовольняє груповим вла-
стивостями і простого, і чистого зсуву, в той час як деформація Hencky не задо-
вольняє груповим властивостям простого зсуву. На основі цього зроблено висновок
про непридатність деформації Hencky в якості еквівалентної деформації простого
зсуву. Висловлено й обґрунтовано гіпотезу про причину відсутності зміцнення ме-
талів при простому зсуві великої величини, яка пояснює, чому коректне (у рамках
механіки кінцевих деформацій) виключення повороту, що відповідний кінцевому
зсуву, призводить до невірного результату.
Ключові слова: простий зсув, інтенсивна пластична деформація, еквівалентна де-
формація, ультрадрібнозернисті матеріали
1. Introduction
Fueled by advances in severe plastic deformation [1], there has been much re-
cent discussion about computing the equivalent strain e in simple shear. Since the
increment of equivalent strain can be defined via the increment of plastic work,
the authors of [2–5] argue for using the von Mises strain
M 3
e γ
= , (1)
Физика и техника высоких давлений 2015, том 25, № 1–2
91
where γ is the shear strain. On the other hand, Onaka [6,7] argues that one must
use the Hencky relation
1/ 22
H
2 ln 1
4 23
e
⎡ ⎤⎛ ⎞γ γ⎢ ⎥= + +⎜ ⎟⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦
. (2)
The justification given by Onaka is the need to exclude rotations from the defor-
mation-gradient tensor in large simple shear deformations.
There is another line of reasoning that has led some authors (see, e.g., [8]) to
formula (2). Starting with Ludwik [9], it has been widely believed that the same
value of e guarantees essentially the same level of hardening, regardless of the
deformation mode. Seen in this light, the estimate (1) is unreasonably high if one
compares the hardening achieved by simple shear and that achieved by other de-
formation modes (e.g., elongation).
In [10], the equivalent strain is defined as a measure of hardening. This
definition gives an expression for the equivalent strain in simple shear pre-
dicting that e saturates at large values of γ. This saturation is in agreement with
an empirically observed fact that, in large deformations caused by high pres-
sure torsion, materials reach a stationary structure without hardening [11–13].
When viewed as a measure of hardening, the equivalent strain becomes a
physical (instead of a geometric) parameter that depends on the material’s
properties.
Here we take a synergetics approach [14], viewing the equivalent strain as an
external control parameter of the deformation process. Using group-theoretic
properties of geometric transformations [15] and the principle of additivity, we
show that the equivalent strain of simple shear must be linear in γ. This approach
gives evidence in favor of using the von Mises strain for computing the equivalent
strain in simple shear.
The paper also proposes a hypothesis explaining the absence of metal
hardening in large deformations caused by simple shear. This hypothesis ex-
plains why excluding rotations as prescribed by the finite strain theory is not
valid.
2. The von Mises strain and group theoretic properties of simple and pure
shear
We view the equivalent strain as an external scalar control parameter deter-
mining the deformation process at low homologous temperatures. We will show
that, for simple shear, such a control parameter must be a linear function in shear
strain γ. For this, consider simple shear as a geometric transformation defined by
relation
=x SX , (3)
where X and x are the initial and the final coordinates of a point, respectively; and
Физика и техника высоких давлений 2015, том 25, № 1–2
92
1 0
0 1 0
0 0 1
γ⎡ ⎤
⎢ ⎥= ⎢ ⎥
⎢ ⎥⎣ ⎦
S (4)
is the transformation operator.
It is easily verified that the set of all simple shear transformations along a fixed
shear direction satisfies the four axioms of group theory [15]. Indeed, consider
two consecutive transformations (4) with shear strains γ1 and γ2 respectively:
( )
1
1 1
1 0
0 1 0
0 0 1
γ⎡ ⎤
⎢ ⎥= = γ⎢ ⎥
⎢ ⎥⎣ ⎦
S S and ( )
2
2 2
1 0
0 1 0
0 0 1
γ⎡ ⎤
⎢ ⎥= = γ⎢ ⎥
⎢ ⎥⎣ ⎦
S S . (5)
It is clear that
( )
2 1 1 2
2 1 1 2
1 0 1 0 1 0
0 1 0 0 1 0 0 1 0
0 0 1 0 0 1 0 0 1
γ γ γ + γ⎡ ⎤ ⎡ ⎤ ⎡ ⎤
⎢ ⎥ ⎢ ⎥ ⎢ ⎥= = = γ + γ⎢ ⎥ ⎢ ⎥ ⎢ ⎥
⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦
S S S . (6)
Thus a composition of two simple shear transformations along the same direction
is a simple shear transformation along that direction, and the set defined by (4)
satisfies the axiom of closure. Similarly, it is easy to see that (4) satisfies the asso-
ciativity axiom, (S3(S2S1) = (S3S2)S1), contains the identity element I = S(0), and
that for every transformation S(γ) it contains the inverse S–1(γ) = S(–γ) such that
S–1(γ)S(γ) = I.
We will associate each transformation S(γ) with the value of its characteristic
control parameter, the equivalent strain es = es(γ), where the subscript «s» stands
for «simple shear». Consider two consecutive simple shear transformations S(γ1)
and S(γ2) with corresponding equivalent strains
( )1 1s se e= γ and ( )2 2s se e= γ . (7)
It follows from (6) that the total shear is characterized by the equivalent strain
( )1 2s se eΣ = γ + γ . (8)
On the other hand, the equivalent strain is additive [16]; this means that
1 2s s se e eΣ = + . (9)
Plugging in expressions (7) and (8), we get
( ) ( ) ( )1 2 1 2s s se e eγ + γ = γ + γ . (10)
A solution to this functional equation is a linear function (see, e.g., [17]):
se C= γ , (11)
for some constant C.
Физика и техника высоких давлений 2015, том 25, № 1–2
93
To match the von Mises values at low equivalent strain [16], ( ) M 3se e γ
γ = =
yielding 1
3
C = . The derivation of relation (11) does not require γ to be small,
and so 1
3se = γ is valid for any value of γ, agreeing with equation (1).
This linear dependence of the equivalent strain from γ follows from group-
theoretic properties of simple shear and the additivity of the equivalent strain. A
similar argument gives a logarithmic dependence in the case of constant-volume
pure shear. Indeed, the transformation operator in this case has the form
( ) 1
0 0
0 0
0 0 1
−
λ⎡ ⎤
⎢ ⎥
λ = λ⎢ ⎥
⎢ ⎥
⎣ ⎦
P , (12)
where λ is the relative elongation.
One can easily verify that the set of pure shear transformations forms a group
whose elements satisfy the relation
( ) ( ) ( )2 1 2 1λ λ = λ λP P P . (13)
Unlike the analogous relation for simple shear (6), it is nonlinear in the sense that
the parameter of the total transformation is the product instead of the sum of the
two components. It is this difference in group-theoretic properties of simple and
pure shear that leads to a different dependence of the equivalent strain from γ in
these processes.
It follows from relation (13) and the additivity property that the equivalent
strain ep (where p refers to «pure shear») satisfies equation
( ) ( ) ( )1 2 1 2p p pe e eλ + λ = λ λ (14)
instead of equation (10) as in the case of simple shear.
A solution of this functional equation is the logarithm function (see, e.g., [17]):
( ) lnpe Kλ = λ , (15)
where K is a constant.
This dependence of the equivalent strain on the relative elongation corresponds
to an expression for the von Mises strain, which is determined by the following
formula [16] in the case of pure shear
( )M
2 ln
3
e λ = λ . (16)
In this case, the von Mises strain follows from group-theoretic properties and the
additivity principle again.
Физика и техника высоких давлений 2015, том 25, № 1–2
94
3. Discussion
The previous section argued for using the von Mises strain in both pure and
simple shear, without relying on any model of the material and without making
any assumptions on the shear strain. Thus the von Mises strain can be used as a
control parameter not only in the case of isotropic bodies where deM defines the
incremental work, but also in other settings – for example, in anisotropic bodies
[16] or in the case of strain gradient plasticity [18].
Two questions arise in light of Section 2 and the discussion [2–8]:
1. Why does strain hardening in simple shear differ substantially from that in
pure shear, for the same value of the von Mises strain, when the von Mises strain
becomes sufficiently large?
2. Why does excluding rotations, as dictated by the finite strain theory, lead to
a wrong result?
To answer these questions, we will first try to answer the contrapositive of
question 1: a polycrystal specimen represents a system with multitudinous degrees
of freedom. So why does a single scalar control parameter determine hardening
and the average grain size (at relatively small equivalent strains and low homolo-
gous temperatures)? This is surprising, like other examples of simple behavior in
complex systems, such as turbulent flows in liquids.
Kolmogorov [19] hypothesized the structural self-similarity of turbulent flows
and its defining scaling laws. Barenblatt [20] gave other examples of simple be-
havior of complex systems related to their self-similarity. We believe that the an-
swer to the contrapositive lies precisely in the self-similarity of metal structure
evolution during deformation. Such self-similarity has been observed experimen-
tally for sufficiently large strains (see, for example, [21]). It is also the self-
similarity that is responsible for the power low
( )f eσ = (17)
of the equivalent stress σ from the equivalent strain e, repeatedly confirmed in
experiments (see, e.g., [16]). In other words, the stress-strain curve is an expres-
sion of scaling behavior, which is common to all self-similar processes according
to [20].
We will show that f(e) is an power low during the self-similar stage of micro-
structure evolution. Consider three consecutive states of the system, specified by
( )1 1;e σ , ( )2 2;e σ , and ( )3 3;e σ . Since the evolution of metal structure is self-
similar, the estimate (17) is invariant with respect to the choice of unit for meas-
uring e. Let us first choose e1 as the unit measure. Then f(e) must have the fol-
lowing form:
( ) 1
1
ef e
e
⎛ ⎞
= σ ϕ⎜ ⎟
⎝ ⎠
, (18)
where ϕ(1) = 1.
For states ( )2 2;e σ and ( )3 3;e σ , according to (18), we get
Физика и техника высоких давлений 2015, том 25, № 1–2
95
2 2
1 1
e
e
⎛ ⎞σ
= ϕ⎜ ⎟σ ⎝ ⎠
, (19)
3 3
1 1
e
e
⎛ ⎞σ
= ϕ⎜ ⎟σ ⎝ ⎠
, (20)
If we choose e2 as our unit measure, a similar argument will lead to the following
relation
3 3
2 2
e
e
⎛ ⎞σ
= ϕ⎜ ⎟σ ⎝ ⎠
. (21)
Multiplying (19) and (21) and comparing with (20), we get
3 3 2
1 2 1
e e e
e e e
⎛ ⎞ ⎛ ⎞ ⎛ ⎞
ϕ = ϕ ϕ⎜ ⎟ ⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠ ⎝ ⎠
. (22)
This implies that ϕ(x) satisfies the following equation
( ) ( ) ( )1 2 1 2x x x xϕ = ϕ ϕ , (23)
A solution to such an equation is a power law (see, for example, [17]):
( ) nx xϕ = , (24)
where n is a parameter. Thus
nAeσ = , (25)
where A is a parameter.
Grain refinement can be viewed as recursive grain subdivision [22]. Kol-
mogorov [23] investigated a fairly general model of recursive particle subdivision,
showing that self-similar structures emerged at a certain stage of this process,
when the subdivision mechanism was constant and scale-invariant. Other research
[21,24–26] showed the universality and scale-invariance of grain refinement dur-
ing plastic deformation. This fact leads to the conclusion that as long as the de-
formation mechanism is unchanged, the metal structure will evolve in a self-
similar manner, resulting in a universal power low stress-strain curve. Thus a de-
viation from this curve must be related to a change in the deformation mechanism.
Papers [10,27] make a case for a hypothesis that a certain percolation mecha-
nism sets in during simple shear. This mechanism explains the lack of deforma-
tion hardening (question 1 above) and the formation of a stationary microstructure
observed experimentally [11–13], as well as a number of other phenomena in
large simple shear deformations [28].
We believe, the answer to question 2 is related to the same percolation mecha-
nism. According to [27], this mechanism is of relaxation nature, periodically re-
lieving internal stress via small rotations of grain clusters. This process allows the
material to repair its structure. This way, the deformed material guarantees a sta-
tionary character of transformation (4) expressed using property (6). Large simple
Физика и техника высоких давлений 2015, том 25, № 1–2
96
shear deformations are realized as a sum of small independent steps, explaining
the special character of rotations in this setting.
4. Conclusion
Simple and pure shear deformations form two groups of geometric transforma-
tions with different properties. The von Mises strain satisfies group-theoretic
properties of both simple and pure shear. The Hencky strain, on the other hand,
does not satisfy the properties of the simple shear group. This brings evidence that
the von Mises strain is the correct measure of the equivalent strain in simple
shear.
Since the von Mises strain is justified using group-theoretic arguments, without
replying on any model of the material, this points to the applicability of the von
Mises strain as a control parameter not only in isotropic bodies but in other set-
tings as well.
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28. Y. Beygelzimer, Mater. Sci. Forum 683, 213 (2011).
Я. Бейгельзимер
ЭКВИВАЛЕНТНАЯ ДЕФОРМАЦИЯ ПРИ ПРОСТОМ СДВИГЕ
МЕТАЛЛОВ
Показано, что простой и чистый сдвиги образуют две группы преобразований с
разными свойствами. Рассмотрена эквивалентная деформация как управляющий
параметр процесса деформирования. Показано, что деформация Von Mises удовле-
творяет групповым свойствам и простого, и чистого сдвигов, в то время как дефор-
мация Hencky не удовлетворяет групповым свойствам простого сдвига. На основе
этого сделан вывод о непригодности деформации Hencky в качестве меры эквива-
лентной деформации простого сдвига. Высказана и обоснована гипотеза о причине
отсутствия упрочнения металлов при простом сдвиге большой величины, которая
объясняет, почему корректное (в рамках механики конечных деформаций) исклю-
чение поворота, соответствующего конечному сдвигу, приводит к неверному ре-
зультату.
Ключевые слова: простой сдвиг, интенсивная пластическая деформация, эквива-
лентная деформация, ультрамелкозернистые материалы
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