ACTA MECHANICA SINICA (English Series), Vol.18, No.5, October 2002
The Chinese Society of Theoretical and Applied Mechanics
Chinese Journal of Mechanics Press, Beijing, China
Allerton Press, INC., New York, U.S.A.
ISSN 0567-7718
NUMERICAL SIMULATION FOR DEFORMATION
N A N O - G R A I N E D METALS*
OF
Yang Wei (~N :P_) Hong Wei ($J~, p~)
(Department of Engineering Mechanics, Tsinghua University, Beijing 100084, China)
A B S T R A C T : Electro-deposition technique is capable of producing nano-gralned
bulk copper specimens that exhibit superplastic extensibility at room temperature.
Metals of such small grain sizes deform by grains sliding, with little distortion occurring in the grain cores. Accommodation mechanisms such as grain boundary diffusion,
sliding and grain rotation control the kinetics of the process. Actual deformation minimizes the plastic dissipation and stored strain energy for representative steps of grain
neighbor switching. Numerical simulations based on these principles are discussed in
this paper.
K E Y W O R D S : nano-grained metals, grain boundary sliding, grain rotation, stress
strain relation
1 INTRODUCTION
When a grain shrinks below a critical size of dislocation depletion that ranges from
15 nm to 20 n m f o r various metals, the available experimental evidence suggests that dislocations seldom present within~ the grain [1~3]. When dislocations were observed, they were
primarily in grains of larger sizes or in immobile or locked configuration. The mobile dislocations are pulled from the grains by image forces that exist in finite atomic ensembles and
overwhelm the dislocation gliding resistance when the grain sizes are small [4].
Nano-grained metals exhibit peculiar deformation behavior. Molecular dynamics simulation [5] indicated that copper of an average grain size of 5.2 n m deformed mainly by many
small, independent slip events in the grain boundaries. T h o u g h dislocations were seldom
observed within the grains, noticeable traces of stacking faults were observed due to the
passage of partial dislocations across the grains. Absence of interior dislocations does not
rule out the capability of plastic deformation of nano-grained metals. Instead, material
scientists had great expectation[ 6] on the super-ductility behavior of nano-gralned materials at room temperature. Recently, Lu et al. [7] reported a stunning experiment of rolling
nano-crystalline copper (about 20 nm grain size) at room temperature (0.23 of homologous
temperature) to an elongation of 5 100%. The specimen was highly purified (99.993 atomic
% of copper) and fully dense (99.4% of the theoretical density of copper) devoid of the artifacts such as imperfect bonding, porosity, contamination and large microstraln that may
trigger the damage localization leading to a premature failure. No texture seemed to form
Received 4 November 2001
* The project supported by the National Natural Science Foundation of China (19972031)
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Yang & Hong: Deformation of Nano-Grained Metals
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during rolling, and the deformed sample at 5 100% elongation exhibited random orientations
of the equiaxed nano-grains, with about the same grain size and slightly increased dislocation density as that of the as-deposited sample. Microstructural analysis suggested that
the superplastic extensibility of the nano-crystalline copper originated from a deformation
mechanism dominated by grain boundary activities rather than lattice dislocation. The bulk
specimen reported in the testing of Lu et al.[7] was synthesized by electro-deposition. Other
techniques, such as ball milling, alloying, consolidation and severe plastic deformation, can
be used to form nano-grained metals. A small grain size is responsible for the unexpected
features in the deformation of sub-micron metals. The role of configuration entropy may be
one of them. It was shown [8'9] that the incorporation of the configuration entropy softened
the stress strain response in a sub-micron regime.
When the grain sizes fall within the range of about I0 nm, it may be envisioned that
crystallized grain cores are glued together by viscous interlayers with atoms of short-range
order. That analogy is similar to the picture depicted by Rachinger [I~ as "oil-emulsion", by
McLean [II] as "ice flows", and by Gifkins [12] as "a rigid core surrounded by a plastic mantle".
Two differences are: (I)for the case of nano-grained metals, the volume fraction of viscous
interlayers is not infinitesimal, and increases as the mean grain size decreases[4'5]; and (2)
the crystallized cores can change the shape with relative ease since neither the ordering nor
the disordering of the atoms involve long range mass diffusion, as suggested by the molecular
dynamics simulation of Schiotz et al.[5]~
Nano-grain metals of such small grain sizes deform mainly by neighboring grains sliding
with insignificant distortion in the grain shape. The crystallized cores move in trajectories
of quasi-unform deformation superimposed by oscillations when they switch the neighbors.
Ashby and Verrall [13] gave a vivid description of the deformation of a four-grain cluster,
where the inter-grain distance (measured from their mass centers) in the elongation direction
increases, and that normal to the elongation decreases. Beside the change in the inter-grain
distance, a rotation of the interigrain link may deem to be necessary.
The distortion and rotation of the neighboring grain cores brings about the issue of
their accommodation. Various accommodation mechanisms include rotation of grain cores,
as well as grain boundary sliding and viscous flow of interlayers to accommodate
the new
locations of the crystalline cores. They control the kinetics of the process. Deficiency in the
accommodating
process leads to local stresses and even the damage nucleation.
The present paper aims at exploring the mechanics framework under the above-mentioned scenario of nano-grains sliding. Sections 2 to 4 describe the formulation on the
kinematic, energetic and kinetic aspects of a meso-scale cell obtained by assembling nanograins. An upper-bound optimization is sought among "kinematically admissible states" by
minimizing the stored strain energy and the plastic dissipation. The geometrically necessary dislocations can be annihilated via the kinetics of grain boundary diffusion. Section
5 describes the numerical simulation in details and discusses the preliminary simulating
results.
2 KINEMATICS
Consider an aggregate of nano-grains that carries an overall finite deformation in terms
of the deformation gradient F~j. For simplicity, consider a plane strain model, such as that
adopted by Lu et al.[7] where little change in the specimen width was reported normal to the
508
ACTA MECHANICA SINICA (English Series)
2002
rolling direction. Bear in mind t h a t the actual three-dimensional response of the aggregate
would be more compliant and accommodating when the plane strain constraint is removed.
Consider a meso-scale element as shown in Fig.1. T h e aggregate in the element consists
of N grains, each has a reference mass center location X ~, and currently occupies a mass
center location x ~, ~ = 1 , . . . , N. Each grain has an effective diameter D% The a - t h grain
m a y undergo a mass translation at the center u s -- x ~ - X ~ and a rotation 0~; it moves
at a velocity v ~ = dx~/dt and spins at an angular speed oa~ = dO~/dt. A set GB is defined
as the pairs of grains that currently share the same grain boundaries. There are about 3N
number pairs (a, fl) belonging to this set. The set evolves as the grain switches its neighbors
during the deformation. The grain boundary shared by the c~-th and the fl-th grains is
denoted by P ~ .
i
--
I i ~._ ~
"T-
---.-- -- -- I
,. . . . . . .
I
I
"
"....
,';
/
i I
i I
.....
I"- /
""" . . . . . . . ' 1
,.-'"1
i
!
I
I
i
I
i
J""
"
"'-i
[ .......
"'"'-/"
I
Fig.1 Grain boundary sliding and grain rotation
Regardless the engaging/disengaging history, the amount of grain boundary sliding
along F ~ is given by
a~--
2
1
D,~ + Da l(u'~--u~) x (x'~--xZ)l + 5 (O'~D'~ +O~DZ )
(1)
In the absence of any accommodation processes, the amount of distortion (in terms of
gapping or overlapping) along the viscous interlayer aligned with F a~ is given by
A~
2
1
n , + D~ l(u"--u~) " (x"-- a~)[ + ~ (O"D" --OaDO) ~
(2)
where ~ denotes a local coordinate along LPa~ with the center of F ~ as its origin.
The overall deformation of the aggregate should be consistent to the prescribed deformation gradient Fij. We impose the overall strain as an affine deformation superimposed
with an oscillation
x~ = F~jx? + w?
(a)
The first t e r m represents the "quasi-uniform" deformation that results in almost parallel
trajectories for the centers of nano-grain cores. Several existing models, such as NabarroHerring creep and Coble creep[ 14] fall in this category. Absence of the second fluctuating
Vol.18, No.5
Yang & Hong: Deformation of Nano-Grained Metals
509
term implies that (a) the grains elongate as the overall deformation in the specimen; and
(b) grains which are nearest neighbors remain nearest neighbors; and (c) the number of
grains across the cross-section of the specimen remains constant [13]. The deformation of
nano-crystalline copper by Lu et al.[7] clearly suggested that it was not the case.
The non-uniform movement w a and the grain rotation 0~ are introduced to complement the quasi-uniform deformation. They are oscillating quantities of null statistic sums.
Under proportional loading whose history is labeled by a strain parameter s, one has
N
w a ( s + As) ----w~(s)
0a(s + As) = 0~@)
E
N
w~ = 0
c~=l
E
0~ = 0
(4)
~=1
where As denotes the strain for a self-repeating step in the deformation process. The
fluctuation feature of the grain rotation was suggested by the scratch mark offset records [15]
and in situ surface experiments [16] concerning superplastic deformation.
The perception of geometrically necessary dislocations (GND) for a polycrystalline
metal [lr] may be borrowed to simulate the integrity of interlayers separating the nanograin cores. In the original formulation of Ashby[ 17], the density of grain boundary GND
consists of two parts: the initial GND density due to the misalignment A0 of neighboring
grains; and the GND density induced by plastic deformation. Since the grain boundaries
after superplastic deformation typically exhibit high angle feature [4,r'ls,191, the dislocation
simulation for the misalignment along the grain boundaries becomes questionable. Thus,
only the interlayer distortion shown in (2) is represented. It may be regarded as three
"geometrically necessary super-dislocations" : two edge super-dislocations of equal amplitude
but opposite signs located at both ends of F ~5, and one edge super-dislocation at the center
of F ~5. The density of the former dislocation pairs, pn, is given by
1
P~= ~
Z
(u~-u~)
"~
(5)
~,flEGB
where n ~/~ denotes the normal of F ~ , S the area occupied by the cell, and b the amplitude
of Burgers vector. The density of dislocations due to grain rotation Pr is given by
1
=
1
I os -
a,/~cGB
where l ~ denotes the length of F ~ .
3 ENERGETICS
3.1 Strain Energy Stored in Geometrically Necessary Dislocations
Geometrically necessary dislocations store elastic energy. One may calculate the strain
energy from the interlayer distortion (2) along F ~p. The first and the second terms on the
right hand side of (2) lead to stress and strain fields symmetric and skew-symmetric in ~,
respectively. The difference in symmetries eliminates their energy interaction, The strain
energy due to the first term in (2)is [2~
En-
(l_v)
Z
a,fl6GB
In
+
(7)
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2002
where G denotes the shear modulus, u the Poisson's ratio, and r0 the cut-off core of the
dislocation. The strain energy due to the second term in (2) can be calculated by the energy
of a tilt grain boundary [21]
Gb
E~-47r(l_u)
l~lO~-:llnlo~-I 0:~1
~ , ~Ee G B
(8)
The formula is obtained by calculating the strain energy for a tile of edge dislocations of
spacing 4/(V~p~D)
3.2 E n e r g y S t o r e d in D i s o r d e r e d I n t e r l a y e r s
Denote Ag as the difference between the specific free energy of the short range ordered
matter and that of the crystallized grain core. It is a positive value. Above a ground energy
state for the crystallized matters, extra energy stored in disordered interlayer is
3N
Ag
E~=T Z ~l~
(9)
~,~6GB
where 6' ~ is the thickness of F ~ .
3.3 P l a s t i c D i s s i p a t i o n
Grain boundary sliding dissipates plastic work. The rate of plastic dissipation per unit
area can be written as
wslido = ~
E
~ (0" - : )
(:
-
t ~ + - -2~
:)
+
J
l~
(10)
a,fl6GB
where D ~ denotes the size of the a-th grain and t~ the tangent of F a~. The grain boundary
sliding resistance TB (0a -- 8 ;3) depends on the current misalignment (@a _ 0~)between the
a - t h and the /9-th grains. Typically, the grain boundary sliding resistance decreases for
higher angle grain boundary.
Plastic dissipation is also induced by the change of interface area. The surplus interface
energy is dissipated as heat so that the process is irreversible[ 13]. Denote 7 ~ as the surface
tension of the grain boundary segment F ~ , one obtains
1
[~/interface = ~
E
"/a~Alafl
(11)
a,fl6GB
Of course one has to consider the dissipation due to mass diffusion. It can be calculated
a~
":~o~on = ~1 ~
l~ ~
(i21
a,/96GB
where J denotes the mass flux and B the mobility.
3.4 E n e r g y M i n i m i z a t i o n
The perturbation in energy due to kinematically admissible evolution can be summarized as
A U - ~ / k E n 3c A E r ~-
AWdisorder ~- AWslid e "~ AWinterface -~- AWdiffusion
(13)
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Yang & Hong: Deformation of Nano-Grained Metals
511
An energetically favorable deformation mode for the nano-grained metal should satisfy the
following conditions:
(1) The strain energy for the geometrically necessary dislocations, plus the accumulated
plastic dissipation by grain boundary sliding, should be minimized.
(2) The minimized strain energy and plastic dissipation should be compensated by the
work done under the prescribed loading against the strain eij of the aggregate.
The first condition can be stated mathematically a s
m i n A U ( Ax a, AO c*, ,'XI ~j, A y aj)
(14)
s.t. Ar
= constant
while the second condition is a statement of the principle of virtual work, as can be phrased
as
m i n A U ( A u ~, A0 ~, A I ~j, A y aj) =- ~
crij/keijdS
(15)
4 KINETICS
4.1 Ordering/Disordering
The ordering or disordering of atoms along the perimeter of a nano-crystalline core
does not involve mass transport. Like an evaporation/condensation process, it was driven
by the difference in free energies of ordered and disordered states, the pressure and the
surface curvature. With the small change of "interface tension" between two phases of long
and short ranged orders omitted, the normal movement of the core perimeter, rn obeys the
relation
= --too-DAy
(]6)
where mo-D denotes the mobility of the ordering/disordering process.
4.2 Grain Boundary and Matrix Diffusion
The atoms within the polycrystal may flow under the gradient of the chemical potential
# under the following Einstein diffusion equation
D
J - /2KB------TV/z
where/2
absolute
l via the
then the
(17)
denotes the atom Volume, D the diffusivity, KB the Boltzmann constant, and T the
temperature in Kelvin. Consider mass flowing from a source to a sink of a distance
lattice and the grain boundaries. We adopt the estimate by Ashby and Verrall[ 13],
atom flowing rate within a grain is
_
1
~ A#~ (Dvl~ + DBh) D
~2KB T ~=1 ~
(18)
where n denotes the number of the source/sink pairs within a grain, and the subscript i
labels the quantities associated with a specific source/sink pair. The quantity A# denotes
the drop in chemical potential from the source to the sink, Dv the diffusivity through lattice
of long-range order, and DB that through viscous interlayer of a short-range order.
ACTA MECHANICA SINICA (English Series)
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2002
The grain boundary flux Ja/~ is driven by the gradient of the stress normal to the grain
boundary
J~3 -
DS~3 0a'~3
KBT Ol
T h a t flux will narrow the gapping or overlapping along F ~
O A~(G t)=
(19)
a s [22'23]
-/20J~;(/'
t)
(20)
For the distortion in grain boundary layer shown in (2), the super GND pair located at the
end of F a~ is relaxed mainly by the normal stress difference of F ~ from the neighboring
grain segments, and the super GND located at the center of F ~ is relaxed by the normal
stress gradient along F ~ .
5 SIMULATION
5.1 G r a i n A s s e m b l y
Two assumptions have been made to form a discrete model of the grains and interlayers:
one is that the interlayer between two grains is a straight-line segment, and keeps straight
while the overall sample deforms; the other is that only triple junctions exist, while four
or more grains could never share a common point. Thus we could interpolate the grain
boundary flux into quadratic functions over each interlayer segment, and rewrite the energy
expressions in terms of nodal flux and the velocity of grains' rigid motion.
Minimizing the total energy, one could abtain nodal flux and grain movement at a
specific instant. The velocity of every triple junction is calculated, and the evolution process
is simulated through integration of the velocity.
5.2 N u m e r i c a l R e s u l t s
A calculation has been made to simulate the evolution process of a cluster of 4 • 4 grains,
which are initially arranged in a uniform hexagon assembly, and subjected to a uniaxial
tension. The grain size is taken as 30 nm. The material constants of copper are employed.
Under periodical displacement boundary conditions, the symmetry of this configuration
prevents any movement or rotation beyond the homogeneous deformation. Figure 2 shows
snap shots of the deformation process. As the elongation proceeds, the assembly shifts from
an armchair configuration to a zigzag configuration.
(
(
(
(
)
)
)
)
Fig.2 Snapshots of deformed assembly of 16 grains starting from a regular
hexagon configuration. From the left to the right, the array shifts from
an armchair configuration to a zigzag configuration
Vo1.18, No.5
Yang & Hong: Deformation of Nano-Grained Metals
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As the cluster elongates, the stress
250
first declines steadily, until reaching a min200
i m u m corresponding to the turning point of
a rhombus assembly, then rises as the as~ 150
sembly evolves to a zigzag hexagon assembly.
~ 100
The simulation clearly supports the AshbyVerrall mechanism [13] at the initial stage of
50
the simulation, but then it is controlled by a
0
mechanism similar to Coble creep [14]. Figure
0.0 0.1 0.2 0.3 0.4 0.5 0.6
3 shows the stress-strain curve of the 16-grain
strain
cluster.
A randomized configuration should
Fig.3 Stress-strain curve of the 16bring out inhomogeneous movements of
grain cluster starting from a regular hexagon configuration
grains. Figures 4 and 5 show the results. The
average radius of grains remains the same as t h a t of the uniform grains. The cluster is
loaded by prescribed displacements. The same material constants of copper are used. As
the sample is strained, the stress strain curve suffers several oscillations, due to the local
switch from an armchair to a zigzag configuration. The stress fluctuates near a value of
about 80 MPa, in fairly close agreement with the threshold stress tested by Cai et al.[24].
i
t=O:O0
t=5:33
t=16:40
t=22:13
l
)
t=11:06
t=27:46
Fig.4 Snapshots of randomly arranged grains
Cases of traction boundary conditions have also been calculated. Figure 6 shows the
evolution of grains that start from the same configuration as Fig.5. The traction boundary condition seems to promote a s y m m e t r y of the cluster. Another issue associated with the
ACTA MECHANICA SINICA (English Series)
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2002
200
180
160
~ 140
12o
"~ 6O
4O
2O
O
9
,
9
i
9
,
9
,
9
,
,
,
9
,
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
strain
Fig.5 Stress-strain curve of randomly arranged grains
t=O:O0
t=25:00
t=8:20
t=33:20
t=16:40
t=41:40
t=50:O0
Fig.6 Snapshots of randomly arranged grains subject to traction boundary conditions
traction boundary condition is the edge effects.
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