1267
The self-similarity theory of high pressure torsion
Yan Beygelzimer
1,2
, Roman Kulagin
*3
, Laszlo S. Toth
1,4
and Yulia Ivanisenko
3
Full Research Paper Open Access
Address:
1
Laboratory of Excellence on Design of Alloy Metals for low-mAss
Structures (DAMAS), Université de Lorraine, Île du Saulcy, Metz,
F-57045, France,
2
Donetsk Institute for Physics and Engineering
named after O.O. Galkin, National Academy of Sciences of Ukraine,
pr. Nauki 46, Kyiv 03028, Ukraine,
3
Institute of Nanotechnology (INT),
Karlsruhe Institute of Technology (KIT),
Hermann-von-Helmholtz-Platz 1, Eggenstein-Leopoldshafen, 76344,
Germany and
4
Laboratoire d’Etude des Microstructures et de
Mécanique des Matériaux (LEM3), Université de Lorraine, UMR 7239,
Metz, F-57045, France
Email:
Roman Kulagin
*
- roman.kulagin@kit.edu
* Corresponding author
Keywords:
deformation mechanisms; high pressure torsion; nanocrystalline
metals; self-similarity; severe plastic deformation
Beilstein J. Nanotechnol. 2016, 7, 1267–1277.
doi:10.3762/bjnano.7.117
Received: 15 June 2016
Accepted: 25 August 2016
Published: 07 September 2016
This article is part of the Thematic Series "Advances in nanomaterials II".
Guest Editor: H. Hahn
© 2016 Beygelzimer et al.; licensee Beilstein-Institut.
License and terms: see end of document.
Abstract
By analyzing the problem of high pressure torsion (HPT) in the rigid plastic formulation, we show that the power hardening law of
plastically deformed materials leads to self-similarity of HPT, admitting a simple mathematical description of the process. The anal-
ysis shows that the main parameters of HPT are proportional to β
q
, with β being the angle of the anvil rotation. The meaning of the
parameter q is: q = 0 for velocity and strain rate, q = 1 for shear strain and von Mises strain, q = n for stress, pressure and torque (n
is the exponent of a power hardening law). We conclude that if the hardening law is a power law in a rotation interval β, self-simi-
lar regimes can emerge in HPT if the friction with the lateral wall of the die is not too high. In these intervals a simple mathemat-
ical description can be applied based on self-similarity. Outside these ranges, the plasticity problem still has to be solved for each
value of β. The results obtained have important practical implications for the proper design and analysis of HPT experiments.
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Introduction
High pressure torsion (HPT) is a severe plastic deformation
process, which is widely used for producing nanocrystalline
metals and alloys [1-3]. The generally accepted theory of HPT
is based on the assumptions of uniformity of simple shear defor-
mation along the height of the specimen and that there is no
slippage between the sample and the anvils. This theory gives a
simple expression for the shear strain
(1)
where β is the torsion angle of the anvil, r is the radial position
and H is the height of the disc.
However, a number of recent experiments and numerical simu-
lations show that the true plastic flow during HPT can differ
Beilstein J. Nanotechnol. 2016, 7, 1267–1277.
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significantly from the theoretical predictions given by the
simple scheme above. In particular, in [4,5] a problem of
coupled phase transformations and plastic flows under torsion at
high pressure in a rotational diamond anvil cell was investigat-
ed. It has been shown that not only the stress, but the strain state
of the sample are strongly dependent on the rheological proper-
ties of the material. In addition, it has been shown that the
assumption of no slippage between the sample and the anvils is
too simple in some cases, and leads to significant errors in the
description of phase transformations. A correct approach to take
into account the slippage in finite element simulations has been
developed in [4,5]. A simple analytical model taking into
account the slippage has been offered in [6]. It leads to an equa-
tion similar to Equation 1 for shear strain with a factor decreas-
ing the angle of rotation of the sample due to slippage. The
same result was obtained experimentally in [7].
The effect of the elasticity of the anvils on the geometry of the
sample and the distribution of the shear strain has been investi-
gated in [8-10]. In particular, it was shown in [10] that the sam-
ples are further deformed during unloading of the anvils and
this produces a peak in the strain in the central region of the
disc.
Finite element modeling is used in [11,12] to show variations of
the stress–strain state along the height of the specimen. In par-
ticular strong shear strain localization near the contact surface
between sample and anvil was quantified in [12]. In [13] a dead
zone is found along the border of the specimen. A double-swirl
pattern in the plane perpendicular to the torsion axis is experi-
mentally shown in [14].
The above results point out the fact that plastic flow during
HPT may have a fairly complex behavior, which should be
taken into account when investigating materials obtained by this
method. Thus, the following question can be raised: Under what
conditions can HPT be modeled in a simple manner?
We will call an HPT process simple if the shear strain in every
point of the specimen is a linear function of the torsion angle of
the anvil, i.e.,
(2)
where Φ(r,z) is a differentiable function of r and z, and z is the
coordinate along the specimen axis as shown below in Figure 1.
This relationship generalizes Equation 1 by inheriting its main
property: the separation between the dependency on β and the
dependency on r and z. According to Equation 2, once we have
Φ(r,z), we have a tool for calculating γ at every point of the
specimen, for any value of the torsion angle. This is exactly
what makes HPT simple. If, on the other hand, one cannot sepa-
rate the β-dependence from the dependence on the coordinates,
i.e., γ = Γ(r,z,β), then, in order to compute γ, one has to solve a
plasticity problem for every angle β. This is what makes the
process complex in terms of its mathematical description.
The question what makes a complex physical process simple to
describe emerges naturally, and the answer often has to do with
the self-similarity of the process [15]. Self-similarity makes a
complex process simple to describe because the process just
repeats itself at different scales of time or space. For example,
the self-similarity of powerful blast waves has allowed
G. I. Taylor to predict damage from nuclear explosions even
before such damage was observed for the first time. These
results were obtained at the beginning of 1941 but were declas-
sified only in 1950 [16]. Many other examples of simple de-
scriptions of complex processes based on their self-similarity
are given in [15].
In this work we investigate plastic flow during constrained
HPT. We show that, within the rigid plastic framework, the
problem we are considering has a self-similar solution if the
deformed material satisfies a power hardening law. In this case,
the shear strain is a linear function of the torsion angle of the
anvil and can be described by Equation 2. We analyze different
properties of self-similar regimes of HPT, and show that the
self-similarity of HPT at the macro-level is connected to the
self-similarity of the structure of the material at the micro-level
hypothesized in [17].
Model
1 Rigid plastic flow formulation for HPT
Figure 1 schematically shows the constrained HPT process [2].
Figure 1: Schematic geometry of the constrained HPT process (the
notation is defined in the main text).
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Within the rigid plastic flow framework [18], the definition of
the stress–strain state (SSS) during HPT is reduced to solving a
set of equations in the cylindrical coordinate system, r, φ, z (see
Figure 1), including (a) the equilibrium equations obtained from
div(σ) = 0:
(3)
(b) the relations between the components of the strain rate
tensor and the velocity vector:
(4)
(c) the von Mises plasticity condition:
(5)
(d) the associated flow rule:
(6)
and (e) the constant volume condition:
(7)
where σ
ij
and with i = r,φ,z; j = r,φ,z are the components of
the stress and strain rate tensors respectively; ν
i
with i = r,φ,z
are the components of the velocity vector; σ = 1/3·σ
ij
·δ
ij
is the
hydrostatic stress,
is the equivalent stress, is the von Mises
strain rate, and σ
s
(e
M
) is the flow stress of the material that
depends on the von Mises strain:
(8)
The set of Equations 3–7 is solved under the following bound-
ary conditions:
(9)
where m is the friction coefficient and ω is the torsion rate of
the anvil.
Under the terms of Equation 9 we believe there is no slippage in
the contact between the sample and the surface planes of the
anvils. According to the experiments in [7] this can always be
achieved by applying a sufficiently high pressure. In subsection
2, we will show that under certain conditions, the HPT problem
has a self-similar solution both with and without slippage.
The torsion angle β does not explicitly appear in Equations 3–7
but the problem is time-dependent because the flow stress σ
s
depends on the von Mises strain e
M
, which in turn increases
with β according to Equation 8.
The global solution of Equations 4–9 has the following form:
(10)
where x is the position vector.
2 A self-similar solution of the rigid plastic
flow problem for HPT
Let y = f(x,t) describe the spatial distribution of a quantity y as a
function of time t, where x is the spatial coordinate. The process
is called self-similar, if the spatial distribution of y at any time t
can be obtained from a reference solution at time t
0
by a simple
similarity transformation:
(11)
where T(t) is time-dependent.
Thus, the spatial distribution of y varies with time while
remaining always geometrically similar to itself. This defini-
tion generalizes the concept of similarity in geometry where
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two figures are called similar if one can be obtained from the
other by uniformly scaling along all dimensions [15].
It can be shown [15] that a self-similar process satisfies the
following scaling law:
(12)
where k is a parameter.
This scaling law suggests that a self-similar solution to the HPT
problem should have the following form:
(13)
where g and q are parameters to be defined, and and
represent a solution to the problem for a given torsion
angle β
0
.
In the following it will be shown that, under certain conditions,
the relationships in Equation 13 satisfy the set of Equations 3–7
and the boundary conditions in Equation 9, i.e., the HPT prob-
lem has a self-similar solution.
We look for a self-similar velocity field in the following form:
(14)
It can be readily seen that such a velocity field automatically
satisfies the condition of constant volume (Equation 7) and also
the boundary conditions (Equation 9) for v
r
and v
z
. According
to Equation 9, when z = 0, the value of v
φ
does not depend on β.
This implies that q = 0, i.e., a self-similar velocity field under
HPT should not depend on the torsion angle of the anvil. In this
case, the components of the strain-rate tensor have the
following form:
(15)
(16)
(17)
Under these conditions the von Mises strain rate is defined by:
(18)
Inserting this expression into Equation 8 it can be seen that if
HPT is self-similar, the von Mises strain in every point of the
specimen is a linear function of the torsion angle β:
(19)
Now we determine the components of the stress tensor for
self-similar flow. They should satisfy the set of equilibrium
equations (Equation 3), the von Mises plasticity condition
(Equation 5) and the associated flow rule (Equation 6). The
latter, with Equations 15–17, gives:
(20)
(21)
(22)
(23)
It can be readily seen that the right hand sides of Equation 22
and Equation 23 scale with β only when the material hardens
according to a power law (where von Mises strain is a linear
function of β in Equation 19):
(24)
where A and n are parameters. Inserting Equation 24 and Equa-
tion 19 into Equation 22 and Equation 23, we get:
Beilstein J. Nanotechnol. 2016, 7, 1267–1277.
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(25)
(26)
where
(27)
(28)
Comparing Equation 25 and Equation 26 with Equation 13, we
obtain: g = n.
According to Equation 20 and Equation 21, and using the
torsional symmetry (derivatives with respect to φ are 0), Equa-
tion 3a and Equation 3c have the following form:
(29)
Thus, according to Equation 20, the normal stresses do not
depend on the coordinates. We obtain:
(30)
where P is the pressure applied in HPT.
By virtue of the torsional symmetry and using Equation 25 and
Equation 26, Equation 3c becomes:
(31)
Inserting this into Equation 27 and Equation 28, and after some
algebraic manipulations, a second order partial differential
equation is obtained:
(32)
where
Equation 32 is a steady-state diffusion equation [19], where u
represents the concentration, and the function Φ, which depends
on the absolute value of the gradient of the concentration and on
the radius r, serves as the diffusion coefficient. According to
Equation 9, the boundary conditions for Equation 32 are the
following:
(33)
(34)
The equation gives another boundary condition for σ
rφ
when
r = R, 0 < z < H (see Equation 9). After inserting Equation 23
and doing some algebraic operations, we get:
(35)
In [19] it is shown that the problem defined by Equations 32–35
has a unique solution. Thus, if the hardening law has a power
form, the problem of rigid plastic flow in HPT has a self-simi-
lar solution, according to which the following relations hold for
the velocity vector
(36)
for the strain rate tensor
(37)
the von Mises strain
(38)