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2011, Acta Materialia

Acta Materialia

Stress-driven migration of grain boundaries and fracture processes in nanocrystalline ceramics and metals2008 •

Nanoengineering of Structural, Functional and Smart Materials

Multiscale Modeling of Stress Localization and Fracture in Nanocrystalline Metallic Materials2005 •

2014 •

Journal of The Mechanics and Physics of Solids

Special rotational deformation as a toughening mechanism in nanocrystalline solids2010 •

2013 •

In this paper we consider grain boundary sliding as a dominant mechanism of plasticity in nanocrystalline metals. A mechanical model for barrier resistance to sliding has been proposed. Characteristic time of plastic relaxation due to grain boundary sliding has been estimated based on molecular dynamics simulation data. At quasi-static deformation yield strength is determined by barrier resistance, and the predicted threshold stresses are in good agreement with experimental data.

Journal of the European Ceramic Society

Energetic design of grain boundary networks for toughening of nanocrystalline oxides2018 •

Improving the mechanical performance of nanocrystalline functional oxides can have major implications for stability and resilience of battery cathodes, development of reliable nuclear oxide fuels, strong and durable catalytic supports. By combining Monte Carlo simulations, experimental thermodynamics, and in-situ transmission electron microscopy, we demonstrate a novel toughening mechanism based on interplay between the thermo-chemistry of the grain boundaries and crack propagation. By using zirconia as a model material, lan-thanum segregation to the grain boundaries was used to increase the toughness of individual boundaries and simultaneously promote a smoother energy landscape in which cracks experience multiple deflections through the grain boundary network, ultimately improving fracture toughness.

Available online at www.sciencedirect.com
Acta Materialia 59 (2011) 5023–5031
www.elsevier.com/locate/actamat
Eﬀect of cooperative grain boundary sliding and migration
on crack growth in nanocrystalline solids
I.A. Ovid’ko a,b,⇑, A.G. Sheinerman a, E.C. Aifantis c
a
Institute of Problems of Mechanical Engineering, Russian Academy of Sciences, Bolshoj 61, Vasil. Ostrov, St. Petersburg 199178, Russia
b
Department of Mathematics and Mechanics, St. Petersburg State University, St. Petersburg 198504, Russia
c
Aristotle University of Thessaloniki, 54124 Thessaloniki, Greece
Received 3 February 2011; received in revised form 22 April 2011; accepted 25 April 2011
Available online 26 May 2011
Abstract
A new mechanism of fracture toughness enhancement in nanocrystalline metals and ceramics is suggested. The mechanism represents
the cooperative grain boundary (GB) sliding and stress-driven GB migration process near the tips of growing cracks. It is shown that this
mechanism can increase the critical stress intensity factor for crack growth in nanocrystalline materials by a factor of three or more and
thus considerably enhances the fracture toughness of such materials.
Ó 2011 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.
Keywords: Nanocrystalline materials; Cracks; Grain boundary sliding; Grain boundary migration
1. Introduction
Nanocrystalline metals and ceramics exhibit superior
mechanical properties that have attracted a rapidly growing research interest in recent years (see, e.g., original
papers [1–15] and reviews [16–23]). However, in spite of
their excellent mechanical characteristics (in particular,
very high strength and hardness), nanocrystalline solids
often demonstrate a brittle behavior, which severely limits
their practical applications (see, e.g., reviews [16–23] and
book [24]).
At the same time, in some cases, nanocrystalline materials exhibit considerable tensile ductility at room temperature [25–27] or superplasticity at elevated temperatures
[28] as well as signiﬁcant fracture toughness that can be
often higher than that of their polycrystalline or singlecrystalline counterparts [24,29–33]. Although the nature
⇑ Corresponding author at: Institute of Problems of Mechanical
Engineering, Russian Academy of Sciences, Bolshoj 61, Vasil. Ostrov,
St. Petersburg 199178, Russia. Tel.: +7 812 321 4764; fax: +7 812 321
4771.
E-mail address: ovidko@nano.ipme.ru (I.A. Ovid’ko).
of good ductility and toughness of some high-strength
nanocrystalline materials is not yet quite clear, it can be
associated with speciﬁc deformation mechanisms, such as
Coble creep [34], grain rotation and sliding [35–37] that
operate in nanocrystalline solids. In particular, recently,
several speciﬁc deformation mechanisms have been
assumed to be responsible for the toughness enhancement
in nanocrystalline solids. These include Ashby–Verall creep
(carried by intergrain sliding accommodated by grain
boundary (GB) diﬀusion and grain rotations) [7], nanoscale
deformation twinning [38], rotational deformation [39] and
stress-driven migration of GBs [40].
Recently, rapidly growing attention has been paid to the
stress-driven migration of GBs in nanocrystalline materials, which represents both a toughening micromechanism
[40] and a speciﬁc deformation mode [19,38,41–46]. This
twofold role of the stress-driven migration of GBs is indirectly supported by experimental observations [47,48] of
the athermal grain growth in the vicinities of cracks in
nanocrystalline materials.
Another important deformation mechanism in nanocrystalline materials is GB sliding (see, e.g., Refs. [17–20,24]).
Non-accommodated GB sliding in nanocrystalline solids
1359-6454/$36.00 Ó 2011 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.
doi:10.1016/j.actamat.2011.04.056
5024
I.A. Ovid’ko et al. / Acta Materialia 59 (2011) 5023–5031
creates defects (dislocations and disclination dipoles) at triple junctions of GBs, which can initiate the formation of
cracks, resulting in brittle fracture of nanocrystalline solids
[49,50]. At the same time, if GB sliding is eﬀectively accommodated, nanocrystalline solids show enhanced ductility
and/or superplasticity [24,28,51]. Accommodation of GB
sliding can be eﬀectively realized through lattice dislocation
emission from triple junctions or diﬀusion [24,28,51].
Recently, a new way to accommodate GB sliding through
GB migration has been suggested and theoretically analyzed
[52]. The discussed (new) accommodation mode results in
the deformation mechanism that involves the cooperative
GB sliding and stress-driven GB migration. In this case,
defects created by GB sliding are, in part, accommodated
by defects created by GB migration. As a corollary, cooperative GB sliding and migration serves as a special deformation mode enhanced compared to pure GB sliding in
nanocrystalline materials [52]. This view is supported by
numerous experimental observations [19,28,51,53] of
concurrent GB sliding and grain growth occurring in nanocrystalline solids during (super)plastic deformation. In Ref.
[52], cooperative GB sliding and migration was theoretically
described as a deformation mode operating in crack-free
nanocrystalline materials. The main aims of this paper are
(i) to describe operation of the cooperative GB sliding and
migration process near crack tips; and (ii) to theoretically
analyze its eﬀect (associated with the local stress relaxation
near crack tips) on the fracture toughness of nanocrystalline
materials.
2. Cooperative grain boundary migration and sliding near a
crack tip: model
Let us consider the geometric features of cooperative
GB sliding and migration in a deformed nanocrystalline
specimen with a crack (Fig. 1). For deﬁniteness, we focus
our analysis on the situation where the crack is ﬂat, and
the specimen is under a tensile load r0 normal to the crack
plane; that is, a mode I cracking (Fig. 1a). In general, one
can distinguish the following geometric types of cracks in
nanocrystalline materials: intragrain cracks (propagating
mostly inside grains; Fig. 1a) and intergrain or GB cracks
(propagating mostly along GBs; Fig. 2a). The intergranular
fracture (Fig. 2a) tends to dominate in nanocrystalline
materials with ﬁnest grains (see, e.g., Refs. [24,54–56]. In
doing so, crack surfaces are curved with the characteristic
size of “curvature facets” being close to the grain size
(Fig. 2a). Since the grain size is very low in nanocrystalline
materials with ﬁnest grains, the characteristic size of “curvature facets” is commonly much lower than the crack
length (Fig. 2a). As a corollary, in the situations where
the role of cracks as stress sources is of critical importance,
intergrain cracks in nanocrystalline materials with ﬁnest
grains are eﬀectively modeled as ﬂat cracks (Fig. 2). The
role of cracks as stress sources is dominant in initiation
of the cooperative GB sliding and migration process as well
as its eﬀect on the stress relaxation near crack tips. There-
(a)
(b)
(c)
(d)
Fig. 1. Grain boundary deformation processes in nanocrystalline specimen near a crack tip. (a) General view. (b) Initial conﬁguration I of grain
boundaries. (c) Conﬁguration II results from pure grain boundary sliding.
Dipole of disclinations AC is generated due to grain boundary sliding. (d)
Conﬁguration III results from cooperative grain boundary sliding and
migration process. Two disclination dipoles CD and BE are generated due
to this cooperative process.
fore, we will use the discussed approximation (Fig. 2) in
our further analysis of the process and its eﬀects on the
local stress relaxation.
Generally speaking, the approximation illustrated in
Fig. 2 can also be exploited in analysis of the tendency of
a material to exhibit either intergranular or intragranular
fracture behavior. In this case, however, its correctness is
not self-evident from geometry, and it can be used only as
a ﬁrst approximation. With the approximation illustrated
(a)
(b)
Fig. 2. A nanoscopically curved intergrain crack and its ﬂat model. (a) A
curved intergrain crack in a nanocrystalline specimen is modeled as (b) a
ﬂat crack.
5025
I.A. Ovid’ko et al. / Acta Materialia 59 (2011) 5023–5031
in Fig. 2, the diﬀerence in crack growth conditions between
intragrain cracks (Fig. 1a) and intergrain cracks (Fig. 2a) is
only in the value of the surface energy penalty due to
formation of crack surfaces. Growth of cracks in grain
interiors (Fig. 1a) is speciﬁed by the surface energy penalty
ce = c with c being the speciﬁc surface energy. Growth of
intergrain cracks (Fig. 2a) is speciﬁed by the surface energy
penalty ce = c cb/2, where cb is the speciﬁc GB energy
(released when the crack moves along the GB).
According to our estimates based on the approximation
illustrated in Fig. 2 (see Appendix A), growth of GB cracks
in nanocrystalline materials with ﬁnest grains is more preferred than growth of intragrain cracks. This theoretical
statement is in good agreement with the experimentally
documented trend (see, e.g., Refs. [24,54–56]) that intergranular fracture is dominant in nanocrystalline materials
with ﬁnest grains. The agreement in question indirectly
supports use of the approximation (Fig. 2) in our further
examination of the cooperative GB sliding and migration
process as well as its eﬀect on the stress relaxation near
the crack tip in nanocrystalline materials with ﬁnest grains.
Thus, let us consider a nanocrystalline specimen which
contains a ﬂat crack and is under a tensile load r0 normal
to the crack plane (Fig. 1a). The applied load and high
stress concentration near the crack tip can induce both
GB migration and sliding near this tip (Fig. 1). These processes release, in part, the high elastic stresses near the
crack tip and thereby can slow down crack growth. Assuming that the intensity of GB migration and sliding and their
eﬀect on crack growth strongly increase with a decrease of
the distance between the crack tip and GBs involved in
these processes, it is reasonable to believe that the dominant eﬀect of GB migration and sliding processes on crack
propagation may be determined by these processes near the
tip.
In the following, we will focus our consideration on the
case of cooperative GB sliding and migration in the vicinity
of crack tips. The geometry of this deformation mechanism
is schematically presented in Fig. 1. Fig. 1a depicts a twodimensional (2-D) section of a deformed nanocrystalline
specimen. Within the model [52], GB sliding occurs under
the applied shear stress and transforms the initial conﬁguration I of GBs (Fig. 1b) into conﬁguration II (Fig. 1c). GB
sliding is assumed to be accommodated, in part, by emission of lattice dislocations from triple junctions (Fig. 1c).
Besides, following Refs. [49,50], GB sliding results in the
formation of a dipole of wedge disclinations A and C in
conﬁguration II (Fig. 1c) characterized by strengths ± x,
whose magnitude x is equal to the tilt misorientation of
the GB (AB is assumed to be a symmetric tilt boundary).
The disclination dipole AC has an arm (the distance
between the disclinations) equal to the magnitude x of
the relative displacement of grains (Fig. 1c).
We further assume [52] that, in parallel with GB sliding,
stress-driven GB migration occurs as well, so that the stress
ﬁelds of defects created by GB sliding are, in part, accommodated by the defects created by GB migration. In the
case shown in Fig. 1, the migration of the grain boundary
AB into another position DE results in the formation of a
quadrupole of wedge disclinations with the strengths ± x
at the points A, B, D and E [52]. The disclination with
the strength + x appearing at the point A due to GB sliding and the disclination with the strength x appearing at
the same point due to GB migration annihilate. The annihilation results in the disclination conﬁguration shown in
Fig. 1d. In general, the cooperative GB sliding and migration process transforms the initial conﬁguration I (Fig. 1b)
into the ﬁnal conﬁguration III (Fig. 1d). During this processes, in parallel with GB sliding that causes the relative
displacement of grains over the distance x, stress-driven
migration of the vertical GB occurs over the distance y
from its initial position AB to the new position DE
(Fig. 1d). The cooperative GB sliding and migration process leads to the formation of two disclination dipoles
CD and BE (Fig. 1d). The disclination dipole CD of wedge
disclinations is characterized by the strength magnitude x
and the arm |x y|. The disclination dipole BE is characterized by the strength magnitude x and the arm y.
3. Energy characteristics of cooperative grain boundary
migration and sliding process: its effects on critical stress
intensity factor for crack growth in nanocrystalline solids
Let us now consider the eﬀect of the applied tensile load
and a long ﬂat mode I crack on the cooperative GB sliding
and migration process in a nanocrystalline specimen
(Fig. 1). The specimen is supposed to be an elastically isotropic solid characterized by the shear modulus G and Poisson’s ratio m. The vertical GB is assumed to be normal to
the crack growth direction and make an angle u with the
grain boundaries AA1 and BB2 (Fig. 1b). Let the triple
junction A lie at a distance p from the crack tip and the
length of all GBs in the initial state (Fig. 1b) be denoted
as d. To calculate the parameters of the cooperative GB
sliding and migration process, let us ﬁrst calculate the
energy change DW associated with the formation of the disclination conﬁguration shown in Fig. 1d. The energy
change DW can be written as
DW ¼
4
X
j¼1
W D ðrj ; hj Þ þ
j1
4 X
X
j¼1
k¼1
4
X
j¼1
sj W Dr ðrj ; hj Þþ
sj sk W int ðrj ; rk ; hj ; hk Þ Asl
ð1Þ
where (rj, hj) are the coordinates of the jth disclination in the
polar coordinate system with the origin at the crack tip
(j = 1–4; see Fig. 1), and the rest of the symbols are deﬁned
as follows: WD(rj, hj) is the energy of the jth disclination in
the solid with a crack; WDr(rj, hj) is the energy of the interaction between the disclination with the strength + x, lying
in the point (rj, hj), and the stress ﬁeld ril induced by the
applied load near the crack tip; Wint(rj, rk, hj, hk) is the energy of the interaction between the jth and kth disclinations
5026
I.A. Ovid’ko et al. / Acta Materialia 59 (2011) 5023–5031
(in the solid with a crack) assuming that both disclinations
have the strength + x, and Asl is the work of the stress ril
done on GB sliding, which does not account the formation
of disclinations. The parameters sj in Eq. (1) account for the
sign of a speciﬁed disclination and are deﬁned as
s1 = s4 = 1, s2 = s3 = 1. The coordinates (rj, hj) are calculated as follows:
1=2
;
1=2
;
r1 ðyÞ ¼ ðy 2 þ p2 2yp cos uÞ
r2 ðxÞ ¼ ðx2 þ p2 2xp cos uÞ
2
r3 ðyÞ ¼ ðy þ ðp þ dÞ 2yðp þ dÞ cos uÞ1=2 ;
2
r4 ¼ p þ d
h1 ðyÞ ¼ arccosðy sin u=r1 Þ;
h2 ðxÞ ¼ arccosðx sin u=r2 Þ;
h3 ðyÞ ¼ arccosðy sin u=r3 Þ;
h4 ¼ p=2
In Eq. (1), we neglected the resistance to GB sliding associated with both the “friction” of the grain boundary
AA1 and the increase of its length in the course of GB
sliding.
The energy term WD(r, h) can be written as
rx0 y 0 ðx0 Þ ¼
K rI sin h2 ðx0 Þ cos2 ð3h2 ðx0 Þ=2 þ 2uÞ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2 2pr2 ðx0 Þ
With Eq. (6) inserted into Eq. (5), we ﬁnd:
Z d
xK r
sin h2 ðx0 Þ cos2 ð3h2 ðx0 Þ=2 þ 2uÞ 0
pffiffiffiffiffiffiffiffiffiffiffi
Asl ¼ pffiffiIffiffiffi
dx
2 2p 0
r2 ðx0 Þ
ð6Þ
ð7Þ
Thus, we have obtained appropriate expressions for all
the energy terms appearing in Eq. (1) for the total energy
DW. The contour maps DW(x/d, y/d) are shown in
Fig. 3, for the situation with nanocrystalline Ni and
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
br
K rI ¼ K br
4Gce =ð1 mÞ is the critical value
IC , where K IC ¼
of the stress intensity factor in the absence of disclinations
(that is, in the case of brittle fracture). In general, K br
IC
depends on ce, where ce = c in the case of an intragrain
crack, and ce = c cb/2 in the case of a GB crack, with c
and cb being the speciﬁc surface energy and speciﬁc GB
energy, respectively. Fig. 3 presents the contour maps
DW(x/d, y/d), for intragrain cracks. In plotting Fig. 3,
we used the following typical values of parameters of
2 2
d hðr=dÞ
, where h(r/d) is a known function
W D ðr; hÞ ¼ Gx4pð1mÞ
[39]. Similarly, the energy term Wint(rj, rk, hj, hk) can be
written as follows:
W int ðrj ; rk ; hj ; hk Þ ¼
Gx2 d 2 gðrj =d; rk =d; hj ; hk Þ
4pð1 mÞ
where gðrj =d; rk =d; hj ; hk Þ is also a known function [39]. The
energy term W Dr ðr; hÞ follows from the expression [57]
Z r
Dr
W
ðr; hÞ ¼ x
rhh ðr0 ; hÞðr r0 Þdr0
ð2Þ
0
The component rhh(r, h) of the stress ﬁeld ril(r, h) is given
[58] by
rhh ðr; hÞ ¼
K rI cos3 ðh=2Þ
pffiffiffiffiffiffiffi
2pr
(a)
ð3Þ
where KI is the stress intensity factor associated with the
applied load r0. Substitution of Eq. (3) into Eq. (2) yields
W Dr ðr; hÞ ¼
4xK rI r3=2 cos3 ðh=2Þ
pffiffiffiffiffi
3 2p
ð4Þ
The work Asl of the stress ril done on GB sliding is
calculated as the work of the stress ril necessary to transfer
a dislocation with the Burgers vector magnitude x (equal to
the length of GB sliding) across a grain boundary AA1 of
length d, i.e.
Z d
Asl ¼ x
rx0 y 0 ðx0 Þdx0
ð5Þ
(b)
0
where (x0 , y0 ) is the Cartesian coordinate system with the
origin at the crack tip and the x0 -axis parallel to the grain
boundary AA0 (Fig. 1), and the component rx0 y0 (r, h) of
the stress ﬁeld ril(r, h) follows from Ref. [58] as
Fig. 3. Contour maps of the energy change DW associated with the
cooperative grain boundary migration and sliding process (near the tip of
a large mode I crack in nanocrystalline Ni) in the coordinate space
(x/d, y/d), for the case x = 17° (a) and 30° (b). The energy DW is given in
units of 108 J m–1.
I.A. Ovid’ko et al. / Acta Materialia 59 (2011) 5023–5031
nanocrystalline Ni and its structure: G = 73 GPa [59],
m = 0.31 [59], c = 1.725 J m2 [60], u = 2p/3, d = 15 nm,
p = 0, x = 17° (Fig. 3a) and 30° (Fig. 3b). The curves with
arrows in Fig. 3 show the lines of the largest (in magnitude)
energy gradient. As it follows from Fig. 3, if the disclination strength x is not very large (Fig. 3a), the minimum
of DW(x/d, y/d) corresponds to some equilibrium value
x = x0 of the GB sliding length, while the equilibrium value
of the GB migration length y is equal to zero. That is, in the
discussed equilibrium state, GB migration is absent. At the
same time, for large enough values of x (Fig. 3b), the minimum of DW(x/d, y/d) corresponds to some equilibrium
values x = x0 and y = y0 of the lengths of GB sliding and
GB migration, respectively.
According to Fig. 3a, for small values of x, one can suggest the following scenario of the cooperative GB migration and sliding process (with the logical assumption that
the lengths x and y change making the gradient of the
energy DW maximum in magnitude). At the ﬁrst stage of
the process, both the characteristic lengths x and y
increase; that is, GB sliding is accompanied by GB migration. However, when the length x of GB sliding exceeds
some critical value (equal to approximately d/2 in the case
shown in Fig. 3a), the length y of GB migration decreases;
that is, GB migration changes its direction. Eventually,
when the length of GB sliding reaches its equilibrium value
x = x0, the migrating GB returns to its original position,
and thus y = 0. Apparently, such a behavior is associated
with a complicated interplay of the disclinations (that
result from GB sliding and GB migration) with each other
as well as with the stress ﬁeld induced by the applied load
near the crack tip.
The dependences of the parameters x0/d and y0/d on
the disclination strength x are presented in Fig. 4, for the
parameter values speciﬁed above. As is seen in Fig. 4, the
equilibrium length of GB migration is small compared to
the length of GB sliding at the considered values of the
stress intensity factor K rI . At the same time, according to
our numerical analysis in the situation with higher values
of K rI , the diﬀerence between the normalized equilibrium
lengths x0/d and y0/d diminishes, so that the contribution
of GB migration (if such migration occurs) to the hindering
of crack propagation increases. For large enough values of
Fig. 4. Dependences of the normalized equilibrium lengths, x0/d and y0/d,
of grain boundary sliding and migration, respectively (near a crack tip in
nanocrystalline Ni) on disclination strength x.
5027
x, the equilibrium lengths x0 and y0 gradually increase with
decreasing x. Below a critical value of x (x 21°), the
equilibrium length y0 of GB migration becomes equal to
zero, whereas the equilibrium length of GB sliding x0
increases very rapidly with a decrease in x, reaching the
values close to the GB length d.
Now let us consider the eﬀect of disclination conﬁguration, resulting from the cooperative GB migration and sliding, on the fracture toughness of a nanocrystalline solid. To
do so, we will use the standard crack growth criterion [61]
based on the balance between the driving force related to a
decrease in the elastic energy and the hampering force
related to occurrence of a new free surface during crack
growth. In the examined case of the plane strain state, this
criterion is given [61] by
1m 2
ðK I þ K 2II Þ ¼ 2c
ð8Þ
2G
where KI (mode I) and KII (mode II) are the stress intensity
factors for normal (to crack plane) and shear loading,
respectively. In the considered situation where the crack
growth direction is perpendicular to the direction of the
external load, the coeﬃcients KI and KII are given by the
expressions
K I ¼ K rI þ k qI ; K II ¼ k qII
ð9Þ
where k qI and k qII are the stress intensity factors associated
with the internal stresses created by the disclinations
located near the crack tip (Fig. 1).
Within the above macroscopic mechanical description,
the eﬀect of the local plastic ﬂow – the cooperative GB
migration and sliding mechanism resulting in the formation
of wedge disclinations – on crack growth can be accounted
for through the introduction of the critical stress intensity
factor KIC. In this case, the crack is considered as that
propagating under the action of the tensile load perpendicular to the crack growth direction, while the presence of the
disclinations simply changes the value of KIC corresponding to the case of brittle crack propagation. In these circumstances, the critical condition for the crack growth
can be represented as (e.g., Ref. [58]): K rI ¼ K IC .
With substitution of Eq. (9) into Eq. (8) and use of the
critical condition K rI ¼ K IC , one ﬁnds the following expression for KIC [39]:
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
2
q
q
K IC ¼ ðK br
ð10Þ
IC Þ ðk IIC Þ k IC
The quantities in Eq. (10) are deﬁned as follows: K br
IC ¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
4Gc=ð1 mÞ, as above, k qIIC ¼ k qII jK r ¼K IC , and k qIC ¼
I
k qI jK r ¼K IC . It should be noted that the quantities k qIIC and
I
k qIC depend on KIC, and, thus, Eq. (10) provides the appropriate formula for the determination of KIC.
The quantities k qI and k qII appearing in the above expression are given [39] by the following relations:
h pffiffiffiffiffi
i
pffiffiffi
k qI ¼ Gx d f1 ðx; yÞ= 2 2pð1 mÞ ;
h pffiffiffiffiffi
i
pffiffiffi
k qII ¼ Gx d f2 ðx; yÞ= 2 2pð1 mÞ ;
5028
I.A. Ovid’ko et al. / Acta Materialia 59 (2011) 5023–5031
where
f1 ðx; yÞ ¼
4
X
pffiffiffiffi
sk ~rk ½3 cosðhk =2Þ þ cosð3hk =2Þ;
k¼1
4
X
pffiffiffiffi
sk ~rk ½sinðhk =2Þ þ sinð3hk =2Þ;
f2 ðx; yÞ ¼
ð11Þ
k¼1
~rk ¼ rk =d:
Using the above expressions, one can numerically solve
Eq. (10) for K IC in the following way. For a preset value of
K rI , one calculates the energy change DW and the equilibrium GB migration lengths x0 and y0 that correspond to
a minimum of DW. Substituting the obtained values of x0
and y0 into Eq. (11), we deduce the values of k qI and k qII .
Then we compute the quantities k qIC and k qIIC with the
assumption that K rI ¼ K IC . On the next step, we calculate
KIC using Eq. (10) and estimate the diﬀerence between
the so obtained value of KIC and the value of K rI . Then
we vary the preset value of K rI and repeat the above procedure until the value of KIC (given by Eq. (10)) becomes
equal to the value of K rI with suﬃcient accuracy.
In order to estimate the eﬀect of the disclinations produced by the cooperative GB migration and sliding process
(Fig. 1) on crack growth, one should compare the critical
stress intensity factor KIC with the quantity K br
IC . To do
so, we have calculated the values of x0 and y0 as well as
the ratio K IC =K br
IC in the cases of nanocrystalline Ni and
nanocrystalline ceramic 3C–SiC with p = d = 15 nm, various values of x and of the other parameters speciﬁed above
(for Ni, we also employed cb = 0.69 J m2 [60]; for 3C–SiC,
we used [62] G = 217 GPa, m = 0.23, c = 1.5 J m2, and
cb = c/2). In the case of an intragrain crack in Ni, the calculations give the following results. For x = 45°, we obtain
x0/d 0.19, y0/d 0.12, and K IC =K br
IC 1:78; for x = 30°,
we obtain x0/d 0.47, y0/d 0.34, and K IC =K br
IC 2:20;
for x = 15°, we obtain y0 = 0. In this connection, it is
noted that at such small enough values of x, one cannot
ﬁnd the value of KIC using the procedure described above.
In other words, in the case under consideration, Eq. (1) for
KIC has no solution. The reason is that in this case, an
increase in K rI increases the equilibrium GB sliding length
x0, which allows one to increase K rI without inducing crack
propagation. An increase in K rI leads to further increase in
x0 which, in turn, allows to increase K rI , and so on. For a
rough estimate of K IC =K br
IC in this case, we assume that
the length of GB sliding cannot exceed GB length d. Then,
for x = 15° and x0 = d, we have: K IC =K br
IC 2:53.
In the case of a GB crack in nanocrystalline Ni, our calculations give the following results. For x = 45°, we obtain
x0/d 0.17, y0/d 0.10, and K IC =K br
IC 1:76; for x = 30°,
we ﬁnd x0/d 0.40, y0/d 0.27, and K IC =K br
IC 2:17; for
2:70.
As is seen,
x = 15° and x0 = d, we obtain K IC =K br
IC
for the same values of x, the values of the ratio K IC =K br
IC
characterizing a GB crack are very close to those characterizing an intragrain crack.
Fig. 5. Normalized critical stress intensity factor K IC =K br
IC vs. grain size d,
in the case of a grain boundary crack in nanocrystalline Ni and 3C–SiC.
Similarly, in the case of an intragrain crack in nanocrystalline ceramic 3C–SiC, one derives the following results.
For x = 45°, we obtain x0/d 0.10, y0/d 0.06, and
K IC =K br
for x = 30°, we ﬁnd x0/d 0.21,
IC 1:68;
y0/d 0.13, and K IC =K br
IC 1:96; for x = 15° and x0 = d,
we obtain K IC =K br
IC 3:40. For a GB crack in 3C–SiC, we
have K IC =K br
IC 1:63, 1.90 and 3.65, for x = 45°, 30° and
15°, respectively.
Thus, the values of K IC =K br
IC in the cases of GB and intragrain cracks are practically the same (at least, for the cases
of x = 30° and 45°). With these theoretical estimates for
nanocrystalline Ni and 3C–SiC, one can conclude that (a)
conditions for occurrence of the cooperative GB sliding
and migration process near intergrain and intragrain
cracks are very similar; (b) the eﬀects of the cooperative
GB sliding and migration process on the local stress relaxation near tips of intergrain and intragrain cracks are very
similar; and, as a corollary, (c) the eﬀects of the cooperative
GB sliding and migration process on crack growth in the
cases of intergranular and intragrain fracture processes
are very similar.
Now let us consider the eﬀect of grain size on the critical
stress intensity factor KIC. To do so, we calculated the
dependence of K IC =K br
IC on grain size d in the case of a
GB crack in nanocrystalline Ni and 3C–SiC. The dependences are presented in Fig. 5, for x = 30°, p = d and other
parameter values (typical of nanocrystalline Ni and 3C–
SiC) speciﬁed above. Fig. 5 demonstrates that, as the grain
size increases from 10 to 100 nm, the ratio K IC =K br
IC
decreases from 2.23 to 1.77, for Ni, and from 1.99 to
1.64, for 3C–SiC. This tendency allows us to conclude that
the suggested cooperative GB sliding and migration mechanism is most eﬀective in fracture toughness enhancement
in nanocrystalline materials at ﬁnest grain sizes. It is contrasted to the situation with lattice dislocation emission
from crack tips – the conventional toughening mechanism
in metallic materials [63,64] – whose enhancing eﬀect on the
fracture toughness of nanocrystalline metals rapidly
decreases with a decrease in grain size [65].
In the practically interesting case of nanocrystalline metals at ambient temperatures, one can perform a rough estimate of the cutoﬀ in grain size at which the suggested
mechanism of fracture toughness enhancement is essential.
I.A. Ovid’ko et al. / Acta Materialia 59 (2011) 5023–5031
In this case, GB diﬀusion is too slow to signiﬁcantly inﬂuence the crack growth, and thereby athermal toughening
mechanisms dominate. As a consequence, in nanocrystalline metals at ambient temperatures, toughening through
the cooperative GB sliding and migration mechanism competes mainly with the conventional toughening mechanism
through lattice dislocation emission from crack tips. In
order to estimate the discussed cutoﬀ in grain size, we postulate that the cooperative GB sliding and migration mechanism is eﬀective in toughness enhancement, if it increases
the critical stress intensity factor larger than lattice dislocation emission from crack tips. In a ﬁrst approximation, the
values of the fracture toughness associated with the two
athermal deformation/toughening mechanisms (cooperative GB sliding and migration and lattice dislocation emission from crack tips) are compared using the
corresponding normalized critical stress intensity factors.
More precisely, we compare the normalized critical stress
intensity factor K IC =K br
IC , which is plotted in Fig. 4 and corresponds to the case shown in Fig. 1, and the factor
~ IC =K br corresponding to the emission of lattice dislocaK
IC
tions along one slip plane from a crack tip. The latter factor was calculated in Ref. [65]. The calculations [65] in the
cases of Al and a-Fe have demonstrated that, for Al,
~ IC =K br < 2 at d < 40 nm and K
~ IC =K br > 2 at d P 40 nm;
K
IC
IC
br
~
whereas, for a-Fe, K IC =K IC < 2 at d < 20 nm and
~ IC =K br > 2 at d P 20 nm. At the same time, in the case
K
IC
shown Fig. 5, K IC =K br
IC is 2, when d is in the range from
20 to 40 nm; and K IC =K br
IC < 2, when d > 40 nm. Therefore,
one can conclude that, for nanocrystalline metals at ambient temperatures, the cooperative GB sliding and migration
mechanism is eﬀective for fracture toughness enhancement,
if the grain size is smaller that the critical grain size being in
the range from 20 to 40 nm. Also, note that both lattice
dislocation slip and thereby emission of lattice dislocations
from crack tips are suppressed in nanocrystalline ceramics
at ambient temperatures. In these circumstances, the examined cooperative GB sliding and migration mechanism can
play a signiﬁcant role in increasing fracture toughness of
nanocrystalline ceramics at ambient temperatures in a
wider range of grain sizes, compared to the situation with
metals.
Thus, the results of our calculations show that cooperative GB migration and sliding along a single GB can make
the critical stress intensity factor KIC several times larger.
Apparently, cooperative GB migration and sliding along
various GBs can increase the value of KIC much further
and, as a result, may lead to a signiﬁcant increase of fracture toughness, as compared to the case of pure brittle fracture. For example, if one hypothesizes that cooperative GB
migration and sliding can increase the value of KIC by a
factor of 5 (as compared to the case of pure brittle intragrain fracture), they would obtain: K IC ¼ 5K br
IC 4:44
MPa m1/2 in the case of nanocrystalline Ni and
KIC 6.50 MPa m1/2 in the case of nanocrystalline 3C–
SiC. The latter value of KIC for nanocrystalline 3C–SiC is
5029
higher than the typical values (3–4 MPa m1/2) of KIC for
polycrystalline 3C–SiC at room temperature. At the same
time, the obtained value of KIC for nanocrystalline Ni is
still much smaller than typical values of KIC for polycrystalline Ni, which are as large as several tens of MPa m1/2.
However, in combination with other toughening mechanisms (limited dislocation emission from crack tips,
stress-driven GB migration, diﬀusion, etc.), the suggested
mechanism of cooperative GB migration and sliding can
result in good fracture toughness of nanocrystalline materials, documented in several experiments [29–33].
4. Concluding remarks
We have theoretically described the cooperative GB sliding and migration process near crack tips and its eﬀect on
the growth of suﬃciently large cracks in deformed nanocrystalline metals and ceramics. The cooperative GB sliding and migration deformation mechanism is shown to
increase the critical stress intensity factors in nanocrystalline metals and ceramics by several times and, as a result,
it may lead to a signiﬁcant enhancement of fracture toughness of these materials. It is shown that among the two constituents of the examined deformation mechanism (GB
sliding and GB migration), GB sliding plays the main role
in the enhancement of fracture toughness in nanocrystalline solids. GB migration is essential in the accommodation
of GB sliding associated with transfer of high-angle GBs
(with a misorientation angle exceeding 21°) (Fig. 1c and
d) and, therefore, the enhancing eﬀect of the cooperative
GB sliding and migration process on the fracture toughness
increases with rising the fraction of high-angle GBs. In the
case of GB sliding associated with transfer of low-angle
GBs near crack tips, GB sliding can be accommodated
through lattice dislocation emission from triple junctions
or diﬀusion [24,28,51].
In general, several deformation mechanisms – lattice
dislocation slip, GB sliding, stress-driven GB migration as
well as rotational deformation modes and other – can contribute to plastic deformation in nanocrystalline materials
(e.g., Refs. [5,17–20,35–37]) and may thus result in fracture
toughness enhancement of such materials [39,40,65]. The
eﬀect of each mechanism on the fracture toughness of a
nanocrystalline specimen depends on the structure of the
specimen and its loading conditions. It is the eﬀective combined action of various deformation mechanisms at certain
conditions that can provide the experimentally observed
[29–33] high fracture toughness of nanocrystalline metals
and ceramics.
Finally, note that fracture toughness of nanocrystalline
materials can also be analyzed through a mechanism-independent approach based on a theory of nanoelasticity, i.e.
linear elasticity enhanced by the Laplasian of strain or stress
to account for the higher-order deformation gradients
induced by the small volume nanoscale constraints. In
particular, enhancement of fracture toughness of nanocrystalline materials may be drawn from the fact [66,67] that the
5030
I.A. Ovid’ko et al. / Acta Materialia 59 (2011) 5023–5031
aforementioned theory of nanoelasticity produces nonsingular stress and strain distributions at the crack tip and
predicts a maximum stress ahead of it, the value of which
depends on the gradient coeﬃcient of nanoscale internal
length; therefore, a size-dependent critical stress intensity
factor can be determined leading to an enhanced fracture
toughness depending on the value of the relevant internal
length parameter.
Acknowledgements
The work was supported, in part, by the Russian Ministry of Education and Science (Contract 14.740.11.0353),
and the Russian Academy of Sciences Program “Fundamental studies in nanotechnologies and nanomaterials”.
Appendix A
In the ﬁrst approximation illustrated in Fig. 2, growth of
both intragrain and intergranular (GB) brittle cracks is
quantitatively characterized by
pffiffithe
ffiffiffiffiffiffiffiffifficritical
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi value of the
stress intensity factor K br
4Gce =ð1 mÞ. Growth of
IC ¼
cracks in grain interiors (Fig. 1a) is speciﬁed by the surface
energy penalty ce = c with c being the speciﬁc surface
energy. In the ﬁrst approximation (Fig. 2), growth of intergrain cracks (Fig. 2a) is speciﬁed by the surface energy penalty ce = c cb/2, where cb is the speciﬁc GB energy
(released when the crack moves along the GB). In these circumstances, the ratio of the critical stress intensity factor
KIC for an intragrain crack to the same factor KIC for a
GBffiffiffiffiffifficrack
the same length is given as follows:
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiwith
ffiffiffiffi
p
2c=ð2c cb Þ. The ratio under consideration is approximately equal to 1.12 and 1.19, for nanocrystalline Ni and
3C–SiC, respectively. (In our estimates for nanocrystalline
Ni, we used the following typical values of parameters
[59,60]: G = 73 GPa, m = 0.31, c = 1.725 J m2 and
cb = 0.69 J m2. In calculation of the ratio for nanocrystalline 3C–SiC, we used the following typical values of parameters [62]: G = 217 GPa, m = 0.23, c = 1.5 J m2 and cb = c/
2.) That is, the absolute values of KIC for intragrain cracks
in nanocrystalline Ni are 12% higher than those for GB
cracks. In the case of 3C–SiC, the absolute values of KIC
for intragrain cracks are 19% higher than those for GB
cracks.
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