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1. Materials Science and Engineering A322 (2002) 1 – 22
An examination of the constitutive equation for elevated
temperature plasticity
Amiya K. Mukherjee *
Di/ision of Materials Science & Engineering,Department of Chemical Engineering and Materials Science,Uni/ersity of California,Da/is,
CA
95616,USA
Abstract
It was 25 years ago that the symposium on rate processes in plasticity was organized. Since then, advances in transmission
electron microscopy, large-scale computation as well as molecular dynamics simulation, etc. have contributed much to our
understanding of elevated temperature plasticity. The constitutive relation that links the stress–strain rate–grain size–temperature
relation (Mukherjee–Bird–Dorn, MBD correlation) was presented in 1968/1969 to describe the elevated-temperature crystalline
plasticity. This equation has held up well during the intervening quarter of a century. It has been applied to metals, alloys,
intermetallics, ceramics, and tectonic systems, and it has worked equally well. It made the depiction of deformation mechanism
maps in normalized coordinates a reality and provided a rationale for estimating life prediction by giving a quantitative estimate
of the steady-state creep rate in creep damage accumulation relationship. In the case of particle-dispersed systems as well as metal
matrix composites, the introduction of the concept of a threshold stress was a substantial improvement in creep studies. One of
the significant applications of the MBD relation has been in superplasticity. The concept of scaling with either temperature or with
strain rate, inherent in this relationship, seems to be obeyed as long as the rate-controlling mechanism is unchanged. The
application of this relation to high strain-rate superplasticity and also to low-temperature superplasticity has been illustrated.
Experimental data demonstrate that superplasticity of nanocrystalline metals and alloys follows the general trend of the
constitutive relation but with important differences in the level of stress and strain hardening rates. It is shown that in the
nanocrystalline range, molecular dynamics simulation has the potential to yield data on stress–grain size–temperature dependen-
cies at very low grain size ranges where experimentalists cannot conduct their studies yet. © 2002 Elsevier Science B.V. All rights
reserved.
Keywords:Creep; Rate process; Superplasticity
www.elsevier.com/locate/msea
1. Introduction
The proceedings of the symposium on ‘Rate Pro-
cesses in Plastic Deformation of Materials’ were pub-
lished [1] a quarter century ago. At that time, engineers
knew the Norton Equation of creep plasticity. Garafalo
[2] also published his monograph on creep of materials.
However, the understanding of the fundamental mecha-
nism that controls the various creep processes at ele-
vated temperatures was still mostly missing. For load
bearing nominally coarse-grained engineering materials
at not-too-low stresses, the creep process is controlled
by the climb of edge dislocations in the vast majority of
cases. This concept was exemplified by Weertman’s
creep model [3], the forerunner in systematizing the
process of creep in terms of the basic parameters, i.e.
stress, temperature, and strain rate. In solid–solution
alloys, a different type of stress dependence of the
strain rate (at intermediate stresses, due to dragging of
solute atoms atmosphere by dislocations) has been dis-
cussed very adequately by Oikawa and Langdon [30].
There was another important parameter, namely
grain size. Geophysicists have encountered the impor-
tance that grain size plays (and specifically fine grain
size) in experimental investigations in mineralogy and
natural observation in plate tectonics and glaciology.
The time was ripe to put together the various observa-
tions into a systematic framework. The outcome,
spelled out in a series of publications by Mukherjee et
* Fax:[phone].
E-mail address
:[email protected](A.K. Mukherjee).
0921-5093/02/$ - see front matter © 2002 Elsevier Science B.V. All rights reserved.
PII: S0921-5093(01)01115-7

2. A.K.Mukherjee /Materials Science and Engineering A322 (2002) 1–222
al. [4,5], Bird et al. [6], and Mukherjee [7], brought
together in a comparative framework the various rate-
controlling mechanisms for elevated temperature plas-
ticity. The deformation mechanism maps came soon
thereafter (Ashby [8], Frost and Ashby [9], Langdon
and Mohamed [10]). The integration of the strain ob-
tained in the region of primary creep (an important
design criterion in power plants and turbine engines)
with that obtained during secondary creep was given by
Garafalo from empirical consideration [2]. This was put
on afirmer footing by Amin et al. [11] and Ahmadieh
and Mukherjee [12] to describe the stress–tempera-
ture–time correlations for high-temperature creep
curves.
Industry has always sought help from materials sci-
entists in terms of obtaining a handle for life prediction
under creep deformation. The Monkman–Grant Law
[13] has been around since 1956. The Mukherjee–Bird–
Dorn relationship gave a predictive mechanism to tie in
the steady-state creep rate with the creep rupture life in
the context of Monkman–Grant Law. It was also
apparent that the so-called power-law of creep defor-
mation does not hold true as stress increases [14] and,
in fact, breaks down at higher stresses. Asbhy and
Frost had suggested [15] that this may be due to a
transition from obstacle-limited thermally activated dis-
location glide to recovery process dominated disloca-
tion climb mechanism. Arieli and Mukherjee [16]
examined this concept in detail in the context of creep
in several fcc materials with a rigorous quantitative
analysis.
It has been apparent in the studies of creep that in
order to decrease the creep rate (industrially always a
desirable path) one should always try to increase the
activation energy of diffusion (of the controlling spe-
cies) and also put impediments on the path of the
gliding dislocations in the form of dispersions. These
particles ideally should befine, befinely dispersed, have
a high modulus, and be incoherent with respect to the
matrix. Dislocations will have to climb over these parti-
cles at elevated temperatures under the presence of
thermal vacancies and stress. However, it was not rec-
ognized early on that this would also give rise to a
threshold stress due to dislocation–particle interaction.
Once this threshold stress [50] was taken into consider-
ation, the apparently high activation energy and high
stress laws, observed experimentally, could be
rationalized.
The constitutive relation for elevated temperature
plasticity has held up well over the past 30 years. It has
been applied to creep of metals, alloys, intermetallic,
and ceramic systems. Another area where it has enjoyed
very successful application is in superplasticity [17,18].
The scaling law inherent in the Mukherjee–Bird–Dorn
relation [5,6] anticipates both low-temperature super-
plasticity [19] and high strain-rate superplasticity
[20,21]. Both phenomena have now been experimentally
verified. Superplasticity is now being actively investi-
gated in nanocrystalline materials [22,23]. The funda-
mental process of grain-boundary sliding and also
interface diffusional processes that are central to the
rheology of superplasticity have been investigated very
recently using molecular-dynamics simulations [24,25].
This manuscript is not intended to provide a compre-
hensive review or references for elevated temperature
plasticity. Instead, it provides an insight into the salient
developments over the past decades and points out
where the area is headed in the future.
2. Constitutive equation
The constitutive equation for elevated temperature
crystalline plasticity was given by Mukherjee, Bird and
Dorn (MBD) in 1968 and 1969 [4–6]. In generalized
form, this equation can be written as:
/¯
s=A
DGb
kT
/b
d+
p/+
G+
n
(1)
where/¯is the steady-state strain rate,Dis the appropri-
ate diffusivity (lattice or grain boundary),Gis the shear
modulus,bis the Burger vector,kis the Boltzmann
constant,Tis the test temperature,dis the grain size,p
is the grain size exponent,+is the applied stress, andn
is the strain rate sensitivity of theflow stress.
It was established fairly early during creep studies
that the apparent activation energies for high-tempera-
ture creep of metals and dilute solid solutions are
independent of the creep strain and insensitive to the
applied stress; they are in close agreement with the
activation energies of self-diffusion. The agreement be-
comes even better when the temperature dependence of
modulus is taken into account and the true enthalpy of
activation of creep is estimated [5].
The changes in the creep rate with time under con-
stant conditions of the independent variables of stress
and temperature are reflections of substructural
changes that accompany creep. In metals, such sub-
structural changes are insensitive to the temperature
and dependent almost exclusively on the time, and the
stress divided by the modulus of elasticity.
Under steady-state creep, the significant substruc-
tural details remain constant. More creep-resistant sub-
structures are produced by higher stresses. Thus, the
steady-state creep rate depends not only on the stress,
but also on the steady-state substructure produced in
the metal by that stress. The significant microstructural
issues in creep were identified earlier, namely, the dispo-
sition and densities of dislocations and, for those mech-
anisms where it is relevant, the subgrain sizes and the
misorientation of subgrains. However, more needs to be
learned about stress-induced migration of subgrain

3. A.K.Mukherjee /Materials Science and Engineering A322 (2002) 1–22 3
Table 1
Creep mechanisms
AMechanism nDiffusivity,D
Volume by vacancyNabarro–Herring 7 ( b/d)
2
1
exchange
Coble Grain boundary by 50 ( b/d)
3
1
vacancy exchange
Volume by vacancyWeertman climb (FCC) 2.5 ×10
6
4.2–5.5 increasing
with Gb/(SFE)exchange
Volume by vacancyWeertman climb (BCC) /2.5×10
6
variable /4.5 but variable
exchange 4.0+n+7
Weertman climb (HCP) Volume by vacancy /2.5×10
6
3.0+n+5.5
exchange
Chemical 3.0+n+3.5Viscous glide in solid 3
solution alloy interdiffusivity
Volume by vacancy 6.0 +n+8.0Dispersion-strengthened Less than 2.5 ×10
6
decreasing with inter-particle spacing and with
increasing particle heightalloys exchange
Grain boundary bySuperplastic creep /100 (b/d)
2
2
vacancy exchange
Volume by vacancy 1.35×10
−11
1Harper–Dorn in Al single
and polycrystals exchange
boundaries and their contribution to creep rates, as has
been emphasized by Blum [26].
The various creep mechanisms that have been iden-
tified are summarized in Table 1, in the context of the
constitutive relationship depicted by Eq. (1). The vari-
ous diffusion-controlled mechanisms differ from one
another only with regard to (a) the diffusivity,D, that
is involved: (b) the value ofAand the structure factors
that are incorporated in it, and (c) the value ofn. Their
values are given in Table 1.
The above description is based on the existence of a
steady state in the creep deformation process in the
secondary stage. This is definitely observed in many
pure metals and solid solution alloys. However, many
commercial creep-resistant alloys generally derive their
high-temperature strength from the presence of precipi-
tate dispersions, which are often unstable during long-
term creep exposure. It is difficult to define a
‘secondary’stage or steady state for such systems, and
the only quantity that can be reasonably reported is a
minimum creep rate. For such practical alloys of tech-
nological significance that have an evolving microstruc-
ture, the steady-state formalism of Eq. (1) should not
apply in a rigorous sense. Empirical formalism like the
/-projection concept, as advanced by Evans and
Wilshire [42], is more appropriate to describe the creep
deformation for such materials.
3. Experimental correlations
The correlation of the Mukherjee–Bird–Dorn ex-
pression with several fcc metals for dislocation-climb
controlled creep is shown in Fig. 1, taken from Ref. [5].
The creep rates of fcc metals seem to increase as the
stacking fault energy (SFE) increases. Barrett and
Sherby [27] interpreted this as an effect of SFE on the
parameter,A, in Eq. (1). The alternative possibility,
however, is thatAfor climb-controlled creep is a
universal constant and that the remaining parameter,n,
of Eq. (1) increases as the stacking fault energy de-
Fig. 1. Steady-state creep rates of nominally pure fcc metals corre-
lated by Eq. (1).

4. A.K.Mukherjee /Materials Science and Engineering A322 (2002) 1–224
Fig. 2. Steady-state creep rates of binary solid solution alloys and
intermetallic compounds, which creep by viscous glide mechanism.
4. A universal law for transient creep
The correlation demonstrated in Section 3 primarily
connects the stress and temperature to steady-state
creep rates. However, in many service applications (as
in the turbine industry), the dimensional changes that
take place during the transient or primary creep are
extremely important. Hence, there was a need to for-
mulate the time–stress–temperature correlation for
dislocation-climb controlled creep that took into con-
sideration the primary stage.
Normal transient creep rates appear to result from
changes in the substructure from that produced imme-
diately upon stressing to that pertaining to the steady
state. The substructures that are produced upon initial
straining at a creep temperature closely resemble those
that are developed during strain hardening over Stage
III deformation at lower temperatures [31]. Most of
the dislocations are arranged in rough cellular pat-
terns, the walls of which are composed of dislocation
entanglements. During the early part of transient
creep, as a result of the extra degree of freedom re-
sulting from dislocation climb, dislocations rearrange
themselves. Consequently, the entanglements disperse
and adjacent cells coalesce to produce more sharply
delineated subgrains. The dispersal of entanglements,
the build-up of subboundaries, the changes in density
of dislocations, and the alterations in misorientation
across subboundaries are not mutually independent.
Undoubtedly, all substructural changes during high-
temperature transient creep are interrelated and de-
pendent upon the dislocation-climb mechanism. Amin
et al. [11], therefore, suggested that transient, as well
as steady-state, creep is controlled by the rate of
climb of dislocations. Ahmadieh and Mukherjee [12]
assumed that the dispersal of dislocation entangle-
ments during primary creep to follow unimolecular
reaction kinetics with a rate constant that depends on
stress and temperatures in the same way, as does the
secondary creep rate. The analysis shows that the
strain (/) versus the time (t) relation can be repre-
sented by:
/=/
o+/¯
st+(=−1)/K[1−exp(−K/¯
st)] (2)
where/
ois the instantaneous strain on loading,/¯
sthe
secondary creep rate,K/¯
sthe rate constant and=the
ratio of initial to secondary creep rates. The stress–
temperature–time data for high-temperature creep
curves for stainless steel, plain carbon steel, Ag, Cu,
Pt, low carbon Ni, Mo, etc. were favorably correlated
with Eq. (2) where the/¯
s, steady-state creep rate was
estimated from Eq. (1). A typical correlation is shown
in Fig. 3 for Cu and Pt from Ahmadieh et al. (details
in Ref. [12]).
creases [6,7]. It is not possible to choose between the
two conjectures due to a variation in experimental
data. Perhaps, the computer modeling of dislocation
plasticity now underway as part of multiscale model-
ing will shed more light on the kinetics of climb of
separated edge partials as a function of stacking fault
energy.
A similar correlation of solid–solution alloys, where
a dislocation–solute atom interaction produces vis-
cous glide creep withn=3, is shown in Fig. 2 (taken
from Ref. [7]). Any interaction mechanism that can
reduce the rate of gliding dislocations to a value that
is so low that glide becomes slower than dislocation-
climb recovery can produce such creep behavior. This
includes dislocation glide in an ordered microstructure
[28,29]. Langdon and Oikawa [30] have adequately
described the role of different solute atoms in terms
of the interaction energy for such Cottrell locking and
the breakaway phenomenon of the dislocation from
such viscous drag at higher stress. A similar correla-
tion for Nabarro–Herring Creep, solid solution alloys
that obey the dislocation-climb creep and creep of bcc
and hcp metals, has been discussed in Refs. [6,7].

5. A.K.Mukherjee /Materials Science and Engineering A322 (2002) 1–22 5
Fig. 3. The universal creep curve for Cu and Pt. With reference to Eqs. (1) and (2) for Cu:/
T=0.0085,K=277,==3.35,n=4.9,A=3×10
6
;
for Pt:/
T=0.01,K=138.6,==2.39,n=7.0,A=5×10
13
.
5. Deformation mechanism map
The idea of representing the deformation behavior of
materials in a two-dimensional map format wasfirst
put forward by Weertman and Weertman [32]. There
are three basic types of two-dimensional deformation
mechanism maps: stress–temperature map, grain size–
stress map, and grain size–temperature map. The
stress–temperature map, where the modulus-normal-
ized stress is plotted as a function of homologous
temperature at a constant grain size, was originally
proposed by Weertman et al. [32] and later developed
in detail by Ashby [8]. Langdon and Mohamed [10]
developed an alternative stress–temperature map by
plotting the normalized stress against the reciprocal of
the homologous temperature. The advantage of the
latter plots is that for the high–temperature deforma-
tion mechanism, all thefield boundaries are straight
lines, whereas in Frost–Ashby-type maps (ground-
breaking as they were) [9], thefield boundaries between
mechanisms having different temperature dependencies,
are curved and their construction requires compara-
tively extensive calculations. The two-dimensional de-
formation maps offer a simple and powerful tool for
visual presentation of mechanical data. The main draw-
back of these maps is their lack of generality. Many
maps must be constructed even for limited ranges of
stress, temperature and grain size, in order to obtain
the overall picture.
Oikawa [33] and Arieli and Mukherjee [34] presented
three-dimensional deformation mechanism maps that
avoid the difficulties mentioned above. Arieli [35] inves-
tigated the temperature–grain size–stress–temperature
dependence of Zn–22% Al eutectoid alloy over large
ranges of stress, temperature and grain sizes. His pri-
mary experimental data are shown in Fig. 4. At low
stresses, two deformation mechanisms are operating:
Coble creep and Nabarro–Herring creep. At intermedi-
ate stresses, the superplastic creep is rate-controlling,
the datum points falling onto parallel lines depending
on the grain size. At high stresses (dislocation climb),
Fig. 4. A double-logarithmic plot of temperature-normalized strain
rate vs. normalized stress for various grain sizes (d×10
−4
cm):/,
0.8;+, 1.2;=, 1.7;×, 2.0;−, 2.9;
/, 3.5 (Al–Zn eutectoid).

6. A.K.Mukherjee /Materials Science and Engineering A322 (2002) 1–226
Fig. 5. A three-dimensional deformation mechanism map for Zn–Al
eutectoid alloy.
microstructure and remaining creep life periodically.
When adequate information about the operating his-
tory of a particular component is available, data ob-
tained from these checks can be used to make remnant
life prediction. On occasion, it is also possible to design
heat treatments in order to extend the remnant life of
the components.
The tertiary creep behavior of Cr–Mo steels has been
analyzed by Ghosh et al. [36]. These steels (including
that studied in this analysis) contain particles that can
coarsen during high-temperature creep. The rate of
creep damage of such material would, hence, be dic-
tated by how fast cavity-type defects had grown and
also by the rate of coarsening of particles under service
environments. If a reliable estimate can be made of
these rates, then the remnant life of components can be
predicted with some degree of confidence.
If only cavity growth is considered, the rate and
extent of damage are governed by the behavior of the
ligaments between cavities. Combining the concept of
Kachanov [37] and Robotnov [38] with Silcock’s [39]
formulation of the particle-coarsening-induced drop in
strength, Ghosh et al. [36] proposed the following equa-
tions for the tertiary creep rate and creep strain:
/¯
/¯
s
=0.5{[1−(t/t
R)]
(/
S//
R−1)
+[1+A(t/t
R)
Y
]} (3)
and
/
/
R
=
[(t/t
R){1+[A/(1+Y)](t/t
R)
Y
}+(/
R//
S){1−[1−(t/t
R)]
(/S//R)
}]
{[1+A/(1+Y)]+/
R//
S}
(4)
where/¯=creep rate at any instantt,/¯
s=steady-state
creep rate,t
R=time to rupture,/=creep strain at any
instantt,/
R=strain at rupture,/
s=/¯
st
R=constant=
Monkman–Grant parameter,A, andY=constant.
Thefirst term in Eq. (3) corresponds to cavity dam-
age, while the second term involves the coarsening of
particles. Here, the contribution from each factor has
been weighted equally. Also, Eq. (4) was obtained by
integration of Eq. (3).
Four specimens [40] were cut from the Cr–Mo–V
steel pipelines used in a Hungarian power plant that
was in service for more than 17 years. The samples were
separated into two lots. Sample 1 and 2 will be referred
to hereafter as Case A, while samples 3 and 4 will be
referred to as Case B. Case B samples were in service at
540°C for 150 000 h under a nominal stress of 60 MPa.
Case A samples had undergone identical service. After
cutting them out of the pipeline, however, they were
given the following heat treatment. These samples were
oil-quenched to room temperature after 10 min at
1000°C. They were then normalized after 3 h at 750°C.
The heat treatment was designed to take some of the
coarser particles into solution and then repreciptate
them on a muchfiner scale.
all the datum points converge onto a single line regard-
less of the grain size. The constitutive equation for
these four mechanisms with respective values for activa-
tion energy, stress dependence, grain size dependence,
and the parameter,A(Eq. (1)), was established by
Arieli and Mukherjee [34] for this Al–Zn eutectoid
alloy. A three-dimensional map was constructed from
these constitutive equations for four mechanisms with
the criterion of equivalency of strain rates for two
pertinent mechanisms at the transition. This three-di-
mensional deformation map is shown in Fig. 5, and the
details for construction are given in the appendix of
Ref. [34]. Inspection of Fig. 4 reveals that the super-
plastic creep domain is limited, at large grain sizes by
Nabarro–Herring creep and dislocation climb at high
temperature, and by Coble creep and dislocation climb
at low temperatures.
One can envisage a dynamic situation (e.g. turbine
bucket in a gas-turbine engine) where stress, tempera-
ture, and grain size all change with time. If we know
the variations in stress, temperature, and grain size with
time (for thefirst two variables, these are design data,
and the third variable can be determined experimen-
tally), and if we neglect the effects of mechanical and
thermal fatigue, then the variation of strain rate with
time can be followed on the three-dimensional map.
This is important if the creep life is to be evaluated.
Very often, in high-temperature applications, the creep
life of a component is limited by the strain approaching
a value that cannot be tolerated rather than by the
occurrence of fracture during creep.
6. Prediction of remnant creep life
The topic of remnant life prediction is of significant
interest to the power-plant industry. Cr–Mo–V steels
are often used for high-temperature service in such
plants. During service, it is desirable to check their

7. A.K.Mukherjee /Materials Science and Engineering A322 (2002) 1–22 7
All four samples were creep-tested to rupture at
540°C at a stress of 132 MPa. The resulting creep data
are shown in Fig. 6(a). It is obvious that the remaining
creep life of the material has been significantly extended
by the heat treatment. Fig. 6(b) shows data points for
/¯//¯
s(normalized creep rate) vs.t/t
R(normalized time)
for sample 1 and 2 (Case A). Here, the steady-state
creep rate,/¯
s, was estimated from Eq. (1) withp=0 (for
coarse grain sizes) from separate studies not reported
here. Fig. 6(c) shows/¯//¯
R(normalized creep strain)
againstt/t
Rfor samples 3 and 4 (Case B). The details
for obtaining values for the constantsAandYare
given in Ref. [40]. In Fig. 6(b) and (c), contributions
from cavity damages are marked‘K–R’, and that from
particle coarsing as‘Silcock.’In Fig. 6(c), it is interest-
ing to note that specimen number 4, which had a much
shorter creep life than specimen number 3 [see Fig. 6(a)]
and, hence, probably had a higher rate of damage
accumulation, followed the K–R curve (which empha-
sizes damage accumulation) more closely than specimen
number 3, which seems to follow Silcock’s particle-
coarsening curve better.
7. Power-law breakdown
When deformed at elevated temperatures, most
metals and alloys obey a power-law creep, i.e./¯×+
n
,
wherenis the stress sensitivity of the strain rate.
However, as pointed out by Sherby and Burke [14] and
Weertman [41], at stress levels higher than about 5×
10
−4
–10
−3
G, whereGis the shear modulus, the
power law breaks down, and the creep rates are higher
(and rise exponentially) than that predicted from an
extrapolation from lower-stress (power-law) regions.
Sherby and Young [95] suggested that the power-law
breakdown might be associated with two contributing
factors: (a) increased contribution from dislocation pipe
diffusion and (b) excess vacancy generation. Nix and
Ischner [96] have put forward a model based on ther-
mally activated glide in the subgrain interior and diffu-
sion-controlled recovery at the subgrain walls. The total
creep rate is given as an algebraic sum of the individual
creep rate due to glide and the creep rate due to
climb-controlled recovery process.
Ashby and Frost [15] suggested that the power-law
breakdown regime is a transition from a diffusion-lim-
ited dislocation motion to the obstacle-limited glide
regime. Arieli and Mukherjee [16] also adopted a simi-
lar approach, i.e. the power-law breakdown with in-
creasing stress signifies a transition from
diffusion-controlled, dislocation-climb-related creep to
glide-controlled thermally activated dislocation mecha-
nism. Raj and Langdon [89] came to a similar conclu-
sion for creep behavior of copper at intermediate
temperatures. They suggested that high-temperature
Fig. 6. (a) Creep curves at 540°C and 132 MPa for four Cr–Mo–V
steel specimens. (b) Normalized creep rate as a function of normal-
ized time for specimens of Case A. (=) Sample 1, (+) Sample 2, (—)
Eq. (3), (- -) Silcock, (––)K–R. (c) Normalized creep strain as a
function of normalized time for specimens of Case B (=) Sample 3,
(+) Sample 4, (—) Eq. (4), (- -) Silcock, (––)K–R.

8. A.K.Mukherjee /Materials Science and Engineering A322 (2002) 1–228
climb is dominant in the power-law creep region, while
obstacle-controlled glide occurring within the cell interi-
ors is rate-controlling in the power-law breakdown
region.
The microstructure of deformed creep specimens sup-
ports the contention that a change of deformation
mechanism occurs at high stresses (or low tempera-
tures). Along with this change in deformation mecha-
nism, there is a change in the activation energy [93]. Raj
and Freed [43] observed that the deformed microstruc-
tures in metals in the primary creep region show a
remarkable similarity with those formed in the expo-
nential creep regime. This observation is in accord with
the conclusion of Hammad and Nix [31] that, upon
initial straining in creep, the substructures closely re-
semble those that are formed during strain hardening
over Stage III deformation at lower temperatures, i.e.
substructures related to thermally activated glide over-
coming localized obstacles.
Arieli and Mukherjee [16] reanalyzed the published
creep data on Pb, Cu, Al and Ni in the power-law
regime and up to the breakdown stress levels. The fcc
metals deform by dislocation intersection mechanism at
low to intermediate temperatures and by creep by climb
of edge dislocations at elevated temperatures. An ex-
pression for the strain rate in fcc metals, where the
rate-controlling mechanisms were glide dislocations in-
tersecting the dislocation trees of the forest, was derived
[16], using the Seeger approach [44] with Friedel [45]
approximation. The significant localized barrier on the
path of the glide dislocation was the trunk of the forest
dislocation trees. Hence, the width of the localized
barrier was the thickness of the trunk of the tree
dislocation, which in turn depended on the separation
of the partials and, hence, inversely, on the stacking
fault energy.
As mentioned earlier, the power-law breakdown phe-
nomenon was explained as a transition between disloca-
tion-climb creep and thermally activated glide. At
constant stress, both mechanisms can be operative
simultaneously, but the mechanism that causes the
highest strain rate will be dominant over a certain range
of temperatures. Arieli et al. assumed that the transition
would occur during constant temperature creep test at
that stress level where the contribution of the two
mechanisms to the strain rate is equal. Conversely, it
can be said that at a constant applied strain rate, the
transition occurs at that temperature where both mech-
anisms require the same stress to produce deformation
rates equal to the applied strain rate.
The transition then can be expressed by equating the
appropriate strain rate equation for the dislocation
intersection process with that for dislocation-climb
creep, the latter being given by Eq. (1) with the appro-
priate values forA,D, andn. (For the dislocation
intersection process, the investigators used Seeger’s in-
tersection theory, a weak localized barrier approxima-
tion and rectangular force–displacement diagram).
Their analysis (see Ref. [16] for details) gave an explicit
expression for the width of the localized barrier,‘d’.
d=
=kT
M(T/T
M)
Gbn=
0.5×b−
/+
G+
L
s
n
−1
××Q
RT
M(T/T
M)
−ln
−
(2a)
1/3
A
−/L
s
b
+
2/3
D
oGb
kT
M(T/T
M)
/+
G+
n−7/3

(5)
Here,T
Mis the homologous temperature,×is the
relative barrier strength (×=0 for weak obstacles),L
sis
the inter-tree distance,−is the Debye frequency,Ais
the same parameter as in Eq. (1), andD
oandQare the
pre-exponential term and the activation energy for dif-
fusion, respectively, and, hence, are related to diffusiv-
ity,D, in Eq. (1).
In constant temperature creep tests, the power-law
breakdown (and, in this interpretation, the transition
from dislocation climb to thermally activated disloca-
tion glide) was observed to occur at a stress level that
asymptotically reaches a constant value. At different
(but constant) temperature tests, different values can be
observed for such asymptotic transition stress. As one
approaches the transition from low-stress, climb-con-
trolled region, the creep-related structural details at
steady state, i.e.A, are substantially constant. At the
other side of the transition where dislocation-glide be-
comes the dominant mechanism, the essential substruc-
tural parameters areL
s,×andd. Since the obstacle
density was assumed to be constant (i.e. one obstacle
on the average per areaL
s
2on the slip plane), and the
strength of the obstacles,×, is not expected to be very
sensitive to temperature and stress, the significant struc-
ture parameter that will vary with variations in test
temperature and transition stress value is‘d,’the‘width’
of the obstacles.
The variation of the obstacle width,d, for the ther-
mally activated intersection mechanism was estimated
from availablecreepdata on the four chosen fcc metals,
Pb, Cu, Al and Ni near the power-law breakdown
region. The value ofdwas obtained using Eq. (5), and
the values for+(transition) were read off directly from
the original log+−log/¯plot of primary data (details in
Ref. [16]). One of the premises of this analysis is that
the power-law breakdown regime coincides with transi-
tion from a diffusion-limited edge dislocation-climb
mechanism to the obstacle-limited glide mechanism.
Hence, approaching the transition point from creep
regime, and using creep data, one should be able to
estimate the significant mechanistic and structural

9. A.K.Mukherjee /Materials Science and Engineering A322 (2002) 1–22 9
Table 2
Comparison of estimated‘d’value with distance of separation of
partials [16]
TemperatureMetal d
est(cm), Eq.d
cal(cm) fromd
est/d
cal
SFE(5)(K)
1.35×10
−7
Al 2.52 ×10
−8
644 5.3
3.83×10
−7
9.48×10
−8
Cu 4.0773
3.6×10
−7
3.8×10
−8
1073 9.0Ni
370Pb 2.93 ×10
−7
6.32×10
−8
4.6
as 80–100) that extends over several decades of creep
rate. The apparent activation energy is also often high
and can be up to three times the value for activation
energy of diffusion. Most theoretical models attempt to
rationalize these observations by introducing specific
structure-dependent terms including the concept of
threshold stress. Bird et al. [6] analyzed the then avail-
able data from literature in 1969 and suggested that a
number of oxide dispersion-strengthened materials
(ODS) exhibited a stress exponent of 7–8, and the
parameter,A, in Eq. (1) was related [47] to
2
/rwhere
is the interparticle spacing, andris the particle
dimension that the dislocation has to climb. Barrett [48]
was perhaps thefirst to point out that the measured
stress exponent can be lowered by replotting the results
of log (/¯/D) versus log(+−+
o) where+
ois a‘friction’or
‘threshold’stress, andDis the lattice diffusivity.
Lagneborg and Bergman [49] introduced the widely
used method of determining the value of threshold stress
by plotting (/¯)
1/n
versus+andfinding the value of+=+
o
where (/¯)
1/n
is zero by the extrapolation method. Lund
and Nix [50] were thefirst to suggest a physical basis for
the origin of the threshold stress. They concluded that
the high strength of ODS alloys is due to the additive
contribution of two terms, i.e. the stress to produce
creep in the matrix alone (in the absence of dispersoids)
and the stress for dislocations bowing around particles.
The latter is the Orowan bowing stress. The Mukher-
jee–Bird–Dorn expression [5] for power-law creep with
this modification can be written as:
/¯=A
DGb
kT
/+−+
o
G
+
n
(6)
whereAis a mechanism dependent constant,+
ois the
threshold stress, and the other terms are the same as
those defined earlier.
It was suggested by Srolovitz et al. [51] that at
elevated temperatures, the occurrence of slip and diffu-
sion at the particle–matrix interface results in an attrac-
tive interaction between dislocation and particle,
allowing the dislocation core to delocalize and relax at
the interface. An empirical model by Mishra et al. [52]
accepts this concept of attractive particle–dislocation
interaction. In addition, it retains the concept of a
threshold stress. However, it alters the creep kinetics by
modifying the dimensional parameter‘A’in Eq. (6) by
introducing a dispersion parameter, i.e. the interpartial
spacing,. Their expression for steady-state creep rate is
given by:
/¯=A+[exp−(104b/×)]
DGb
kT
/+−+
o
E
+
n
(7)
Mishra, Nandy and Greenwood [53] suggested a physi-
cal basis to explain the attractive dislocation particle
interaction. They presented a model based on the disso-
ciation of lattice dislocations into interfacial disloca-
tions when they enter the matrix–particle interface for
parameters for the obstacle-limited glide mechanism, i.e.
in this case, the parameter‘d.’The estimated‘d’values
in this analysis involving pure fcc metals should then
relate to the width of the repulsive trees in the forest for
dislocation intersection mechanism. The value of‘d’for
the four different fcc metals should also depend on the
equilibrium separation of the partial dislocations as a
function of stacking fault energy. For the prototype case
of edge dislocation, that equilibrium separation can be
obtained from Ref. [46]. Table 2 gives the values for‘d’
estimated from Eq. (5) and also calculated from the
stacking fault energy values. The two values of‘d’are
within one order of magnitude for each metal.
It is appropriate to summarize the assumptions and
approximations that are inherent in this analysis. These
are: (a) assumption of a rectangular force–displacement
diagram; (b) consideration of only random obstacles,
whereas, in fact, they are more likely to be in the form
of cellular or subgrain arrays; (c) consideration of only
moderately weak obstacles; (d) mobile dislocation den-
sity equal to total density; (e) no variation of line energy
of dislocation with orientation; (f) negligible or non-ex-
istent internal back stress; (g) uncertainties in the re-
ported values for SFE that are often measured at
ambient temperature, but were used here at elevated
temperatures; (h) estimation of the separation of partial
dislocations for equilibrium condition neglecting the
obvious local shear stress that will be present in actual
mechanical tests; and (i) difficulty in obtaining the exact
asymptotic transition stress from the double logarithmic
plot of stress–strain rate data on creep from published
literature. However, in spite of these, the approximate
analysis does suggest that the power-law breakdown
may, indeed, be correlated with the onset of a low-tem-
perature, thermally activated glide mechanism, e.g. dis-
location intersection process for the fcc metals that were
investigated here.
8. Creep mechanisms in dispersion-strengthened
materials
The experimental results on creep of a dispersion
hardened system reveal an unusually high stress sensitiv-
ity of creep rate (nwell over 20 and sometimes as high

10. A.K.Mukherjee /Materials Science and Engineering A322 (2002) 1–2210
climb bypass. Their model predicts a threshold stress
for creep that depends on the particle radius, interparti-
cle spacing, and reduction in dislocation self-energy
because of dissociation.
Eq. (6) does not include any dispersion parameter
dependence apart from the presence of a threshold
stress. Fig. 7(a) shows a plot of (/¯kT)/(DGb) against
normalized effective stress (+−+
o)/Efor a number of
dispersion-strengthened alloys with metallic/intermetal-
lic dispersoids. For comparison, the data for pure alu-
minum are also included. If the presence of a threshold
stress was the only manifestation of second-phase dis-
tribution, the data on this plot should have merged, but
it is apparent that the kinetics of creep is influenced as
well. This is to be expected, as the dislocation has to
bypass the particle by local climb. Shewfelt and Brown
[54] have modeled the bypass of particles by local climb
of edge dislocations. They predict that the time for an
edge dislocation to bypass a spherical particle is pro-
portional tor
2
/. Mishra, Paradkar and Rao [52] tried
to correlate the dimensionless constant,A(Eq. (6)), for
the alloys considered here with the parameterr
2
/.
They could not observe any correlation. However, a
correlation can be noticed between‘A’and the parame-
ter (b/)
1/2
, as shown in Fig. 7(b), which is related to
the attractive interaction between dispersoid and dislo-
cation at elevated temperatures. The correlation can be
expressed as:
A=8×10
8
exp[−103.8(b/)
1/2
] (8)
Eq. (8) represents the magnitude of strengthening due
to presence of dispersoids. When the interparticle spac-
ing approaches infinity (i.e. a pure matrix with no
dispersoid), the exponential part becomes unity, and the
constant 8.3×10
8
represents the creep strength of the
pure aluminum matrix. This constant matches quite
well with the value reported by Frost and Ashby [55]
for pure aluminum (takingn=5 andE=3G). Incorpo-
ration of Eq. (8) into Eq. (6) leads to Eq. (7). In a
recent presentation to honor Johannes Weertman on
his 70th birthday, Mukherjee and Mishra [56] reana-
lyzed some of the carefully documented data on ODS
alloys (with different grain sizes of the matrix) as well
as aluminum matrix composites with SiC whisker or
Al
2O
3fiber dispersions. Although other models for such
a dispersion-hardened system exist [57], they concluded
that the steady-state creep behavior of dispersion-hard-
ened system is, indeed, better explained by a power-law
creep with a temperature-dependent threshold stress, as
exemplified by the modified Mukherjee–Bird–Dorn ex-
pression, represented by Eq. (6). Additionally, for
metal-matrix composites, a dislocation creep mecha-
nism map has been proposed [56,94] for depicting the
transition from dislocation–reinforcement phase inter-
action controlled creep (withn=5) to constant sub-
structure creep (withn=8) in a solid solution matrix of
aluminum alloy composites.
9. Superplasticity
Superplasticity is being investigated nowadays both
for its intrinsic merit in the context of fundamentalflow
and failure mechanisms and for its technological signifi-
cance in forming operations. There are considerable
savings in material costs and in labor-intensive machin-
ing costs in this near-net-shape forming process. It is
not the intent to present here an exhaustive review of
the phenomenon of micrograin superplasticity. Several
books and reviews for general reading have appeared
over the past several years [17,18,58,59].
Instead, the current author wishes to focus on the
subareas where rapid progress is being made at the
Fig. 7. (a) Temperature and diffusivity compensated strain rate vs.
normalized effective stress for rapid solidation processed (RSP) alu-
minum alloys. (b) Correlation of the dimensionless constant,A(in
Eq. (8)), with inverse square-root of the interparticle spacing.

11. A.K.Mukherjee /Materials Science and Engineering A322 (2002) 1–22 11
Fig. 8. Normalized strain rate vs. normalized stress for TZP and
TZP/Al
2O
3composite.
9.1.Superplasticity in ceramics and geological
materials
In ceramic materials, crystal bonding is strong and
directional, so motion of dislocations is often a difficult
process. Thus, the structure of the interface is impor-
tant for a fundamental understanding of superplasticity
in ceramics. Carry and Mocellin [61] have emphasized
the extreme sensitivity of grain-boundary diffusivity to
the local chemistry. The importance of deliberate addi-
tion of dopants that can change the viscosity of+-
spodumene glass and thereby increase the superplastic
forming rate was shown by Wang and Raj [62] where
the stress dependence of the strain rate,n, was one.
Such‘engineered’interfaces can be of great advantage
as a processing variable.
Other ceramics, however, often show an‘n’value
equal to 2. Wakai et al. [63] observed that infine-
grained yittria-stablized tetragonal zirconia polycrystals
(YTZP), tested in tensile superplastic deformation, the
n-value was equal to 1, thep-value was equal to 2, and
the value for the activation energy for deformation was
closer to the self-diffusion energy of cation species. The
investigation [64] on superplasticity of TZP/Al
2O
3com-
posite (grain size/0.5=m) revealed trends very similar
to that of Y–TZP alloy, including a stress sensitivity of
2. These experimental results on TZP and TZP/20 wt%
Al
2O
3composite have been plotted in Fig. 8 as diffusiv-
ity normalized strain rate vs. normalized stress. (The
details for the values for the parameters used for nor-
malization are given in Ref. [66]. The deviation of the
data for 1350°C for the TZP/Al
2O
3composite for this
correlation was also evident in the primary experimen-
tal data, and the explanation may lie in the original test
conditions and circumstances).
In retrospect, it appears that there is much that is
common in the description of micrograin superplastic-
ity between metals and ceramics. The basic structural
requirements remain the same, i.e. afine, equiaxed
grain size that is reasonably stable during deformation
and high angle grain boundaries. Several investigators
have commented that whereas in metallic systems, su-
perplastic behavior can be observed for grain sizes that
are typically/10=m, the corresponding grain sizes for
superplastic ceramics are significantlyfiner, usually less
than 1=m. We believe that this is associated with the
scaling law, which is implicit, for high-temperature
plasticity depicted by Eq. (1). It was shown (details in
Ref. [17]) while comparing superplasticity in an alu-
minum-based matrix and that for Y-TZP at 0.75 of
their respective melting point that:D
gb
AL/D
gb
Zr
4+
=d
AL
/
d
ZrO
2
=10
Since superplasticity is a rate-controlling mechanism
that is encountered at strain rates lower than that for
dislocation creep, and since geological creep rates are
virtually always lower than technologically significant
current time. This includes the area of high strain rate
superplasticity, evidence of superplasticity in geological
materials, and in plate tectonics and nanocrystalline
superplasticity.
Superplasticity is a close cousin of creep, and in fact,
it is one of the various elevated temperature
micromechanisms of deformation. Superplasticity obeys
the general formalism of the correlation depicted by
Eq. (1), with values ofn=2,p=2–3 and an activation
energy equal to usually the activation energy for grain-
boundary diffusion and occasionally equal to that for
volume diffusion.
One of the dominant miscrostructural features in
superplasticity is the role played by grain-boundary
sliding (GBS). The grain compatibility during GBS is
maintained by concurrent accommodation processes,
which may involve grain-boundary migration (GBM),
grain rotation, diffusion or dislocation motion. Most of
the models proposed in the literature (for a review, see
Ref. [17]) for superplasticity generally consider one or
the other accommodation process in conjunction with
GBS. The accommodation mechanisms considered can
be divided into three general groups: (a) diffusional
accommodation, (b) accommodation by dislocation
motion and (c) combined models having elements of
both dislocation and diffusional accommodation. The
rate equations (correlatingflow stress, strain rate, grain
size, and test temperature) derived from different mod-
els have essentially the same form. This is true despite
the differences in the detailed concepts associated with
individual models [60].

12. A.K.Mukherjee /Materials Science and Engineering A322 (2002) 1–2212
creep rates, it would be interesting to explore the exis-
tence of superplasticity in geological materials. The
best-characterized investigation in this area is the work
of Schmid et al. [65] onfine-grained limestone. The
material, Solenhofen limestone (grain size 4=m) was
tested in compression (with a confining pressure) at
temperatures 0.54–0.73T
m. The normalized plot of
their results is shown in Fig. 9. Details of the parame-
ters used to construct this plot are given in Ref. [66]. In
then=5 regime, it was clearly dislocation-climb creep
with an activation energy that was close to that for the
activation energy of diffusion of carbon or oxygen in
calcite. The value of the parameter‘A’in Eq. (1) in this
regime was estimated to be 2×10
7
, which correlates
very well with that given by Mukherjee–Bird–Dorn
[5,7] for dislocation creep. The investigators also ob-
served transient stage and formation of subgrains in
this regime.
At lower stress levels, they observed ann-value of
around 1.7 and an apparent activation energy that was
significantly less than the activation energy of carbon or
oxygen self-diffusion in calcite. The strain rate de-
pended strongly on grain size with the exponent‘p’in
Eq. (1) being equal to 2 or 3. There was no evidence of
subgrains, and the individual grains remained mostly
equiaxed. Grain boundary sliding observed microscopi-
cally accounted for approximately two-thirds of the
strain. In this author’s opinion, this regime is clearly
associated with superplasticity.
Until recently, only dislocation creep and diffusional
creep had been considered in describing theflow behav-
ior of upper mantle. The texture studies do, indeed,
indicate the operation of the dislocation creep in tec-
tonic environments. However, at low stresses and at
higher temperatures, the possibility of superplastic
mechanism cannot be ruled out. Superplasticflow has
already been proposed for olivine-rich mylonites by
Boullier et al. [67]. Superplasticity could conceivably be
a viable deformation mechanism in the upper mantle of
the earth. It may be useful to reanalyze some of the
earlier data that have been discussed in the literature in
the context of‘diffusional creep.’
9.2.High strain-rate superplasticity
During the early years of development of superplas-
ticity, the grain sizes used to be approximately 15=m,
and the optimum strain rate used to be about 10
−4
s
−1
. With the development of processing methods to
produce an ultrafine grain size, it has been possible to
move up the optimum forming rates to approximately
10
−1
–10
−2
s
−1
. In this range, high strain rate super-
plasticity (HSRS) has been explored aggressively in
recent years.
The essence of obtaining HSRS is to refine the grain
size. With reference to Eq. (1), at constant temperature,
decreasing the value of the grain size‘d’will increase
the strain rate. This grain-size refinement is usually
carried out by thermomechanical treatment that pro-
ducesfine, hard, dispersed particles that pin the grain
boundary and retard the tendency of strain enhanced
grain growth during the superplastic forming operation.
In such a microstructure, intragranular particles would
hinder the movement of lattice dislocations from pass-
ing from one side of the grain to the other side, thereby
making the slip-related accommodation step during
superplasticity difficult. Furthermore, the intergranular
particles will directly influence grain-boundary sliding
(GBS) by acting as barriers to sliding. The net result is
the existence of a microstructural threshold stress.
The above concept will be illustrated by analyzing
the HSRS data of several powder metallurgy aluminum
alloys. The constitutive equation for superplasticity has
been modified [68] in the context of the presence of this
threshold stress. The modified expression is the same as
Eq. (6) except that, here, the parameter,D, refers to
grain-boundary diffusion. The data set used in the
present analysis is taken from Ref. [69]. The tempera-
ture-dependent threshold stresses for the individual al-
loy data were estimated using the Lagneborg–Bergman
plot. The details for such estimation, including caution
in the proper use of then-values, etc. are discussed in
Refs. [69,70]. In Fig. 10, the data for a large number of
modified conventional aluminum alloys [69] have been
plotted according to the Eq. (9)
//=40
D
oEB
kT
exp/
−
84,000
RT
+/b
d+
2/+−+
o
E
+
2
(9)
Fig. 9. Normalized strain rate vs. normalized stress for Solenhofen
limestone (grain size 4=m).

13. A.K.Mukherjee /Materials Science and Engineering A322 (2002) 1–22 13
Fig. 10. Variation of grain size and diffusivity normalized strain rate
with normalized effective stress for modified conventional aluminum
alloys. The use of grain-boundary self-diffusivity and a grain-size
dependence of−2 results in the best merger of the data.
grain-boundary diffusion have been predicted by the
models of Mukherjee [71] and Ball and Hutchison [72].
Even the dimensionless constant 40 obtained from the
bestfit over all of the experimental data shown in Fig.
10 agrees well with the predicted dimensionless constant
of 25–50 [73].
It is appropriate to examine the role of the size of
particle located in the grain during GBS. Fig. 11(a)
shows the classical concept of slip accommodation dur-
ing grain-boundary sliding. The particles at grain
boundaries [e.g. particle markedP
gbin Fig. 11(a)]
impede the grain-boundary sliding and lead to stress
concentration. The stress concentration must be low-
ered for continuous sliding to take place and to avoid
cavity formation. The stress relaxation can occur by
diffusionalflow of atoms around particles, as depicted
in Fig. 11(a). If the rate of diffusional relaxation is fast
enough to remove the stress build-up, the overall grain-
boundary sliding will not be influenced parametrically.
Koeller and Raj [74] have pointed out that the stress
concentration at the particle/matrix interface above a
critical strain rate may result in decohesion at that
interface leading to formation of voids and consequent
loss of ductility. The critical strain rate for cavity
nucleation is given by:
/¯
c×C
(1−−)[1−2−+(2/−)]
(5/6−−)
2
/G×
kT+/V
fD
g
d
p
3
+
(10)
where/¯
cis the critical strain rate,−is the Poisson ratio,
Gis the shear modulus,×is the atomic volume,V
fis
the volume fraction of particles,d
pis the particle di-
ameter, andis the grain-boundary width. The numer-
ical constant,C, has a value of 9 (Mori et al. [75]). This
equation is based on relaxation by interfacial diffusion.
It is instructive to evaluate the applicability of the
diffusional relaxation on the onset of HSRS in MA6000
(an ultrafine-grained oxide dispersion strengthened
nickel-based alloy). Taking values of−=0.33,V
f=
0.025,d
p=10 nm,D
v,Gand×from Frost and Ashby
[55],/¯
c=0.3 s
−1
, and the constantC=9 (according to
Mori et al. [75]), the critical temperature at which the
diffusional relaxation rate is fast enough to relax the
stress build-up is found to be/1150 K. The prediction
of Eq. (1) is compared with the data of Singer and
Gessinger [76] for MA6000 alloy in Fig. 11(b). It can be
noted that the increase in observed ductility agrees well
with the threshold for cavitation predicted by the diffu-
sional relaxation model. Hence, the broad aspects of
HSRS can be explained by afine and stable grain size
due to particle pinning of the grain boundaries. HSRS
requires a temperature-dependent threshold stress due
to the presence of the particle, and a critical combina-
tion of temperature and strain rate so that the stress
concentration at the grain-boundary particles due to
GBS can be relaxed by diffusionalflow of atoms
around the particles.
Fig. 11. (a) Schematic of grain-boundary sliding with particles in the
grain boundary and grain interior. (b) Onset of high strain-rate
superplasticity in mechanically alloyed nickel-based alloys as a func-
tion of temperature.
The stress dependence of 2, an inverse grain-size depen-
dence of 2, and an activation energy close to that for

14. A.K.Mukherjee /Materials Science and Engineering A322 (2002) 1–2214
9.3.Nanocrystalline superplasticity
Nanocrystalline materials have a very large density of
grain boundaries. Therefore, they have attracted con-
siderable discussion on the extension of grain-
boundary-related deformation mechanisms such as in
superplasticity. Bulk nanocrystalline materials provide
an opportunity to investigate the scalability of the
grain-size-dependent phenomenon to a muchfiner
scale. Significant levels of superplasticity have been
observed in nanocrystalline Ni, Ni
3Al, and Al–Mg–Li
alloys (Al-1420). Fig. 12 shows deformed tensile sam-
ples of some of these materials. The Ni
3Al and Al-1420
specimens were processed from ingot specimens by
severe plastic deformation (SePD), namely torsion
straining (true strain 6–7) under a high pressure of 1.5
GPa. The nanocrystalline Ni specimens were prepared
by electro-deposition process.
9.3.1.HSRS in nanomaterials
The nanocrystalline 1420 Al was found to be super-
plastic at a strain rate of 10
−1
s
−1
. Table 3 shows that
by refining the grain size from 6=m to 100 nm, the
strain rate for superplasticflow can be increased from
10
−3
to 5×10
−1
s
−1
, i.e. an increase of 500 times.
This was anticipated from Eq. (1) but was only demon-
strated recently [77]. This has a significant impact on
developing superplastic materials.
9.3.2.Low-temperature superplasticity in nanomaterials
One of the impressive and technologically important
observations [19] is low-temperature superplasticity in
nanocrystalline materials. For technological applica-
tions, the reduction of superplastic forming temperature
in titanium alloys and Ni
3Al alloys is desirable. For
example, a Ni
3Al alloy (see Table 4) with a 6=m grain
size shows superplasticity at 1050°C. This temperature
is higher than the current industrial practice of/
900°C as an upper temperature limit for superplastic
forming of superalloys. The decrease in superplastic
temperature (to 725°C in nanocrystalline Ni
3Al alloy)
brings it within the conventional superplastic forming
range, and therefore, current die and tooling methods
can be used.
9.3.3.Extensi/e strain hardening
Extensive strain hardening has been noticed in all
alloys tested so far [22,77,78]. Fig. 13 shows the stress–
strain curve for a 50 nm grain size specimen of Ni
3Al
under superplastic testing conditions. There is some
grain growth during superplastic deformation. Because
of the grain-size dependence of theflow stress at con-
stant temperature and strain rates (see Eq. (1)), strain
hardening during superplasticity has been convention-
ally explained in terms of grain growth. At a constant
temperature and strain rate and forp=2,n=2inEq.
Fig. 12. Tensile superplasticity in nanocrystalline Ni, 1420–Al alloy
and Ni
3Al.
Table 3
HSRS in nanocrystalline Al–Li–Mg Alloy (Al-1420)
100 nm grain size 6=m grain size
(Kaibyshev [58])(present study)
Percentage 400390
elongation
450300Temperature (°C)
1×10
−3
Strain rate (s
−1
)5×10
−1
Table 4
Low-temperature superplasticity in Ni
3Al [77]
50 nm grain size 6=m grain size
560Percentage elongation 560
Temperature (°C) 725 1050
6×10
−4
1×10
−3
Strain rate (s
−1
)
Fig. 13. Comparison offlow behavior of nanocrystalline and micro-
crystalline Ni
3Al alloy. The nanocrystalline alloy exhibits an equiva-
lent ductility at significantly lower temperatures. Note the very high
flow stress and strain hardening in the nanocrystalline specimen.

15. A.K.Mukherjee /Materials Science and Engineering A322 (2002) 1–22 15
(1), the change in stress,+, is expected to be directly
proportional to change in the grain size,d. However,
grain growth alone cannot account for all of the strain
hardening in all these materials. For example, super-
plastic deformation of a Ni
3Al alloy at 650°C and at
1×10
−3
s
−1
strain rate shows aflow stress increase by
a factor of/5, whereas the grain size increased only
by a factor of 2.
9.3.4.High flow stresses
Fig. 13 also shows a comparison offlow curves
during superplasticity in a Ni
3Al alloy. This alloy was
tested both in microcrystalline (d=6=m) and
nanocrystalline states (d=50 nm). The highflow
stresses in the nanocrystalline sample are apparent and
are examined in the next section. Although the test
temperature and grain sizes are different for the two
microstructural conditions, the comparison is interest-
ing because the overall ductilities in both cases are
similar.
In TEM studies of superplastically deformed Ni
3Al,
dislocation activity appeared to be quite limited. A few
lattice dislocations were visible. There was no large-
scale storage of dislocation in the substructure that can
explain the high strain hardening or highflow stresses.
Sometimes, there was evidence [79] that the dislocation
line was associated with grain-boundary sources. This
would be consistent with slip accommodation where the
dislocations are generated not in the grain interior but
at the grain boundary during GBS.
The author and colleagues [86] recently conducted
room-temperature in-situ tensile tests in TEM on speci-
mens of nanocrystalline Ni
3Al. Three observations were
noteworthy:
1. Dislocation [marked‘A’in Fig. 14(a)] is pinned at
opposite grain boundaries and moved through the
grain interior in a manner as if the dislocation
pinning points were dragged along the grain-
boundary plane. This was not an uncommon
observation.
2. After approximately 11% straining, the TEM in-situ
tensile device was stopped, and the deformed speci-
men in the specimen holder was tilted (with respect
to electron beam) to various angles up to 30°and
imaged. Fig. 14(b) shows that tilting altered the
contrast of the grains, as expected, but it did not
reveal any discernible storage of dislocations in the
microstructure. Hence, the high strain hardening in
nanocrystalline Ni
3Al cannot be explained by dislo-
cation storage. (As already stated, it cannot be
entirely explained by strain-enhanced grain growth
either.)
3. A most remarkable three-dimensional sliding and
rotation of the nanocrystalline grains were observed.
With progressive deformation, the extent of sliding
and rotation decreased (from visual observations of
the video image) and eventually stopped in the
region under observation. Occasionally, a crack was
observed to open up in the TEM foil specimen that
then lowered the stress in the region being observed,
and grain-boundary sliding and rotation ceased
there. Although the TEM foil may not fully repre-
sent bulk behavior, the in-situ specimen had three to
four grains in the thickness direction.
The author believes that this in-situ TEM observa-
tion on deformation of nanocrystalline grains may
provide an explanation for the very highflow stresses
that were observed in the stress-strain curve (see Fig.
13) of nanocrystalline Ni
3Al. Zelin et al. [90,91], in their
in-situ tensile deformation in SEM of ultrafine-grained
Pb–Sn eutectic specimen, observed cooperative phe-
nomena at interfaces, such as cooperative sliding of
grain groups, grain group rotation, and cooperative
grain-boundary migration. Often, 30 grains or more
participated in such cooperative grain-boundary sliding
(CGBS). It is interesting to raise the question: How will
CGBS proceed if, in the sequential process of CGBS, a
particular grain boundary had an orientation and struc-
ture that made it very difficult to slide? Kurtz and
Hoagland [92] used a computer simulation to study the
effect of grain-boundary dislocations on the sliding
resistance of−11 grain boundaries in aluminum. For
the case of a pure screw dislocation in the−11 {323}
grain boundary with a Burger vector of (a/2) [101], the
peak stress to start grain-boundary sliding was high, in
the order of 1.4 GPa. It is interesting to note that the
peakflow stress encountered in the superplastic defor-
mation of nanocrystalline Ni
3Al (at 650°Cand/¯=1×
10
−3
s
−1
) can also be high, i.e. 1.5 GPa (see Ref. [22]).
It would not be unreasonable to suggest that the very
highflow stresses encountered in nanocrystalline super-
plasticity may be related to the difficulty to produce
sliding at specific places in the CGBS surfaces, where
the state of the grain-boundary structure and the nature
of the grain-boundary dislocations are such that sliding
is very difficult and can only take place at very high
stress levels.
The reason as to why the microcrystalline nickel
aluminide did not reveal such highflow stresses could
possibly be attributed to the following reason. For a
given specimen thickness, the probability offinding a
grain boundary that is unfavorably oriented for sliding
is less in microcrystalline materials than in nanocrys-
talline materials. In current investigations with tensile
specimens having a 1 mm thickness, the number of
grains in the thickness direction is approximately 1000
times greater in a 50 nm grain size specimen than in a
10=m grain size specimen. Thus, in a typical microcrys-
talline specimen, the probability offinding a grain
boundary that is very difficult to slide would be 10
3
times less than the similar probability in nanocrystalline
specimens. However, the alternate explanation, i.e. that

16. A.K.Mukherjee /Materials Science and Engineering A322 (2002) 1–2216
Fig. 14. (a) Still frame from the video imaging of in-situ tensile deformation in TEM of nanocrystalline Ni
3Al. Dislocation marked‘A’is pinned
at opposite grain boundaries moved through the grain interior in a manner as if the dislocation pinning points were dragged along the
grain-boundary plane. (b) Tilting experiments to different tilt angle of nanocrystalline Ni
3Al deformed in-situ in TEM to approximately 11%
strain. No dislocation storage was observed in the microstructure.
the highflow stresses associated with nanocrystalline
superplasticity can be due to the difficulty of disloca-
tion generation in nanograins or due to the slower
kinetics of diffusion at the much lower test tempera-
tures (where nanograins are relatively stable), cannot be
discounted either. Experiments under way in the au-
thor’s laboratory are expected to shed some light on
this conjecture.
9.3.5.Comparison of superplasticity models and
microcrystalline materials
Fig. 15 shows the variation of normalized stress with
temperature and grain-size-compensated strain rate for
Ti–6Al–4X alloys. The primary data were taken from
Salischev et al. [80] and from the present study. Tita-
nium alloys exhibit slip-accommodated, lattice diffu-
sion-controlled grain-boundary sliding. The data [81]
for microcrystalline Ti–6Al–4V alloy and the expected
behavior from the Sherby–Wadsworth correlation are
also included in thisfigure.
Two observations are apparent from thisfigure.
First, the data for the microcrystalline alloy agree well
with the expected trend for slip-accommodated, lattice
diffusion-controlled grain-boundary sliding. Second,
the data for the submicrocrystalline and nanocrystalline
specimens show a higherflow stress or slower kinetics

17. A.K.Mukherjee /Materials Science and Engineering A322 (2002) 1–22 17
at the same normalizedflow stress. A similar trend was
also observed in a Zn–Al eutectoid [82] where the
superplasticflow is grain-boundary diffusion-con-
trolled. The observed higherflow stress for submicro-
crystalline and nanocrystalline material suggests a
transition in micromechanism or due to additional
difficulty in grain-boundary sliding because of the
difficulty for slip accommodation to occur at these very
small length scales [22].
10. Molecular dynamics simulation — deformation of
nanocrystalline materials
Experimental studies with nanocrystalline material
have been constrained by the limits set by processing
and consolidation techniques. Materials produced by a
severe plastic deformation route normally can produce
grain sizes no smaller than about 30–40 nm. Routes
involving the inert-gas condensation method do give
powder sizes that can be less than 10 nm, but the
consolidation process often introduces porosity gradi-
ents as well as contamination originating from powder
surfaces.
One way to explore the characteristics of deforma-
tion of nanomaterials at truly low length scales (around
10–15 nm or less) is with a molecular-dynamics simula-
tion. It is possible to study high-angle grain-boundary
sliding or diffusion creep in nanocrystalline material in
a region where experimentalists would have great
difficulty conducting their studies. One can try tofind
the grain size below which continuum dislocation pile-
ups do not form [83,84]. Hence, the Hall–Petch Law
(+×d
−0.5
) is expected to be obeyed only above this limit.
Recent investigations on thin-film nanocrystalline
metallic multilayer nanoindentation results of Misra et
al. [84] show that continuum pile-up behavior (+×h
−0.5
)
is observed at a layer thickness, h−/20–50 nm. At
h+10 nm, single dislocation behavior [+×h
−1
ln(h)] is
observed, as expected from Orowan bowing. In the
intermediate range, a modified Hall–Petch behavior
with an exponent different from minus 0.5 may explain
the dependence of+onh. At even lower dimensions in
the Cu–Ni system, the investigators noticed softening
ath+3.5 nm, consistent with the onset of non-disloca-
tion deformation modes. In a recent paper, Misra et al.
[85] have taken such an analysis further by developing
a two-dimensional deformation mechanism map for
such nanocrystalline thin-film multilayers in terms of
grain size and layer thickness. The map delineates the
operative zones for continuum dislocation pile-up (the
Hall–Petch Law is obeyed), discrete pile-up (the
modified Hall–Petch Law is obeyed), single dislocation
regime (Orowan bowing is possible), andfinally at
considerably lower dimensions (less than 3 nm for
Cu–Cr and Cu–Nb multilayers) where no dislocation
activity is possible. They observed quite a good correla-
tion between the experimentally observed grain size vs.
layer thickness and the theoretical predictions. Their
results were supported by atomic-force microscopy ob-
servations of nano-hardness indents where crystallo-
graphic slip traces were observed ath=50 nm, but a
shape change with no clear crystallographic slip traces
was observed ath3.5 nm [86].
One can study the competing plastic deformation
mechanisms in nanophase metals, using molecular dy-
namic simulation. In nanophase solids, depending upon
grain size, as much as 30% of the total atoms can be
boundary atoms. Here, intercrystalline deformation
mechanisms are expected to be relevant and dominant,
and intracrystalline dislocation activity will be difficult
and perhaps impossible. The latter conjecture is based
on the fact that dislocation sources inside grains cannot
possibly exist because of size and image force limita-
tions. One can only envisage dislocations emitted from
a boundary that can eventually travel across the grain.
Van Swygenhoven and Caro [25] performed molecular-
dynamics (MD) simulations of deformation in three-di-
mensional nanophase Ni samples, in the temperature
range 300–500 K at constant applied uniaxial tensile
stress between 0 and 3 GPa in the grain size range
3.2–12 nm. Their simulation cell volume wasfilled with
nanograins grown from seeds with a random location
and random crystallographic orientation,filling the
space according to the Voronoi construction. The sys-
tem was relaxed to give a minimum energy, yielding
nanograins with mainly high-angle grain boundaries.
The details of their methods for the MD simulation are
given in Ref. [25].
Van Swygenhoven et al. analyzed the deformation
results in terms of the grain size, and the evolution of
Fig. 15. Flow stress normalized by modulus, as a function of strain
rate normalized by grain size and diffusivity for Ti–6Al–4X alloys.

18. A.K.Mukherjee /Materials Science and Engineering A322 (2002) 1–2218
Fig. 16. (a) Molecular dynamics simulation of plastic deformation.
Strain–time curves for nanophase Ni between 3.4 and 12 nm grain
size, with high-angle grain boundaries at a stress of 1.5 GPa at 500 K
(results from Van Swygenhoven et al. [87]). (b) Strain rate vs. inverse
of grain size, from Fig. 16(a) (results from Van Swygenhoven et al.
[87]).
energy,+is the stress,/is the activation volume, and
kThas the usual meaning of Boltzmann constant times
absolute temperature. The results analyzed with this
equation, for random high-angle samples, show a well-
characterized behavior of intergrain plasticity in the
lower limit of grain size. There was clearly a softening
as grain size decreased. By further increasing the grain
size, a clearly different behavior was observed [seefig.
16(b) from Ref. [87]], which represents a transition into
a new regime that should lead, at even larger grain sizes
(simulation experiments currently underway), to the
intragrain plastic behavior expected from the Hall–
Petch relation. Their analysis of the simulated data at
temperatures up to 500 K and load, using Eq. (11)
yielded an activation energy,U, of 0.2 eV, comparable
to the self-interstitial diffusion barrier in crystalline Ni.
The corresponding value for the activation volume,/,
was 16D, i.e. in the range of one atomic volume. A
recent investigation by Schonfelder et al. [88] on com-
puter simulations of high-angle grain-boundary migra-
tion (a process that, in many ways, is comparable to
grain-boundary sliding) reported an activation energy
and volume that were in the same range as that ob-
served by these investigators.
Hence, such a MD simulation demonstrates that for
a small grain size (d10 nm), plastic deformation in
nanocrystalline Ni has a regime controlled by grain-
boundary sliding where the strain rate increases with
decreasing grain size. Above a critical grain size in the
range of 10 nm, the strain rate becomes less dependent
on the grain size, indicating a transition into a different
regime. For the grain-size range studied until now by
these investigators, this regime is intermediate between
the sliding regime and the intragrain dislocation-domi-
nated Hall–Petch regime that is typical of coarse-
grained polycrystals. The grain-boundary sliding is
suggested to occur by individual jump events by quite
mobile atoms within the interface, inducing viscous
slide of a grain with respect to each other and (in the
range of conditions studied here) without crack or
cavity formation.
A MD simulation was used for thefirst time by
Keblinski, Wolf, and Gleiter [24] to study diffusion
creep. Working with pure silicon with a carefully cho-
sen interatomic potential, they tailored their input mi-
crostructure to have a uniform grain size and shape and
to contain only high-angle grain boundaries. The uni-
form grain size and identical grain shape were chosen
to eliminate grain growth or at least to suppress it on a
MD time scale (of typically 10
−9
s).
An important assumption in the derivation of diffu-
sional creep formulations is the requirement that the
deformation mechanism be‘homogeneous’, i.e. the
elongation of the grain size not accompanied by sliding.
To assess whether their simulated system satisfied this
basic assumption, Keblinski et al. [24] monitored the
the microstructure in terms of energy, density, and
atomic bond analysis. At the early stages of deforma-
tion at a stress of 1.5 GPa, strain increased linearly with
time in all cases for samples with several grain sizes
[Fig. 16(a), from Ref. [87]]. The number of atoms with
perfect crystal energy only decreases by 2–3% or less
after deformation. From a further analysis of such
data, they concluded that from room temperature up to
500 K, no damage accumulation takes place during
such deformation of grains having dimensions below a
critically small grain-size range. These investigators as-
sumed that macroscopic displacement is the result of
grains sliding against each other with a general non-lin-
ear viscous behavior. No contribution to plasticity from
the grain interior was observed. They suggested a stan-
dard stress-assisted activation process where the strain
rate as a function of grain size, temperature, and stress
is given by:
/¯=
d+
0
d
e
(−U/kT)
sinh
/+v
kT+
(11)
wheredis the grain size,d
0is the strain (due to
intergrain sliding) per unit time,Uis the activation

19. A.K.Mukherjee /Materials Science and Engineering A322 (2002) 1–22 19
position of the centers of mass of the simulated grains
as a function of simulation time. Their observations on
the MD simulation confirmed that the center of mass of
the simulated grains did not move. This attested to the
homogeneous nature of their deformation process, i.e.
absence of grain-boundary sliding in their attempt to
demonstrate pure diffusional creep.
For such a highly idealized, designed microstructure,
Keblinski et al. were able to show that the creep
behavior is controlled by grain-boundary diffusion and
described quantitatively by the expression of Coble
creep (see Table 1 for rate parameters). Specifically,
they found the activation energy for grain-boundary
diffusion creep to be the same as that for grain-
boundary self-diffusion (Fig. 17). They also found that
the strain rate increases with decreasing grain size as
1/d
3
, consistent with Coble creep theory. However, they
noticed a non-linear increase of the strain rate with the
applied stress. This can be explained in terms of the
high stresses that were applied in their simulations so as
to make the effect observable on a MD time scale. At
low stresses, where the stress times atomic volume
kT, the simulated data on creep rate vs. stress can,
indeed, be rationalized with a one-power stress depen-
dence of the creep rate as required by Coble creep.
However, the total creep strains were only 1–2% in
their simulations. Hence, the possibility that both the
microstructure and the atomic structures of the grain
boundaries may change significantly for larger total
strains (i.e. longer deformation times) cannot be
excluded.
10.1.Reflection on MD simulation
Many of the experiments on plastic behavior of
nanocrystalline materials are often conducted using
hardness measurements. These (as well as a few creep
studies) are performed in the temperature regime corre-
sponding to Coble creep and Nabarro–Herring creep.
Such processes involve atomic diffusion and, hence,
take place in a time scale that is large compared to
atomic vibrations. Computer simulation experiments
are restricted to much shorter time scales and, conse-
quently, have to be done at much higher stress levels, in
order to keep the time factor within tractable dimen-
sions. Furthermore, the atomic interaction used within
MD simulations is derived from an empirical model. Its
quantitative prediction capabilities, when extrapolated
far from the region from where it wasfitted, as in the
case of disordered interfaces, have to be considered [25]
with care. However, the technique has the power of
highlighting the physical processes involved, and careful
interpretation of the data can provide a quantitative
evaluation of the parameters controlling the tempera-
ture, stress, and grain-size dependencies, which have
been the central focus of this manuscript. Additionally,
the approach can yield simulated data at a very low
grain size range where experimentalists cannot easily
conduct their studies yet.
The performance of supercomputers is increasing
ever so rapidly in terms of processing speed and mem-
ory. It should be possible in the not-to-distant future to
investigate by MD simulation, in comparatively larger
nanoscale grains, the complex interplay of grain-
boundary sliding, grain-boundary migration and grain
rotation, and grain-boundary diffusion-controlled creep
mechanisms. These are all essential inputs to the under-
standing of superplastic behavior offine-grained mi-
crostructures in metals and ceramics. When that point
is reached, it will no longer be necessary to restrict MD
simulations to idealized and artificially stabilized
nanocrystalline microstructure. Predictions from MD
simulation experiments in nanocrystalline plasticity can
then be compared directly with real-life experiments
and each approach will be able to compliment the
other.
11. Conclusions
Over the last 25 years, advances in materials process-
ing and testing as well as advances in electron mi-
Fig. 17. MD simulation of Coble creep in nanocrystalline Si. (a)
Activation energy for Coble creep; (b) activation energy for grain-
boundary self-diffusion under zero stress (results from Keblinski et al.
[24]).

20. A.K.Mukherjee /Materials Science and Engineering A322 (2002) 1–2220
croscopy, large-scale computation including multiscale
modeling and also molecular dynamics simulation, have
contributed much to our understanding of elevated
temperature plasticity.
The constitutive relation that links the stress–strain
rate–grain size–temperature relation (Mukherjee–
Bird–Dorn, MBD correlation) was presented in 1968–
1969 to describe elevated temperature crystalline
plasticity. This correlation has held up well during the
intervening quarter of a century. In addition to metals
and alloys, it has been applied to intermetallics, ceram-
ics, and tectonic systems, and it has worked equally
well.
This correlation made the depiction of deformation
mechanism maps in normalized coordinates a reality
and provided a rationale for estimating life predictions
by giving a quantitative estimate of the steady-state
creep rate in the creep damage accumulation relation-
ship. A universal creep curve can be constructed from
the steady-state creep rate estimated from the MBD
correlation. The power-law breakdown at high stresses
in creep can be described in terms of a transition from
climb-controlled recovery creep to thermally activated
glide-controlled dislocation motion, overcoming local-
ized obstacles. In the case of particle-dispersed systems
as well as metal–matrix composites, the introduction of
the concept of a threshold stress substantially improved
the creep correlation.
One of the significant applications of MBD relation
has been in superplasticity. The concept of scaling with
either temperature or strain rate, inherent in this rela-
tionship, seems to be obeyed as long as the rate-con-
trolling mechanism in microcrystalline material is
unchanged. The application of this relation to high
strain rate superplasticity and also low-temperature su-
perplasticity has been illustrated. It was shown that
superplasticity of nanocrystalline metals and alloys fol-
lows the general trend of this constitutive relation but
with important differences in the level of stress and
strain hardening rates. In the nanocrystalline range,
molecular-dynamics (MD) simulations can yield simu-
lated data on stress–grain size–temperature dependen-
cies at a very low grain-size range where experimental
work is limited.
Acknowledgements
This manuscript could not have been written without
the contribution of the author’s graduate students and
post-doctoral associates over three and a half decades
—a contribution that is fondly acknowledged. It is a
privilege to acknowledge the support from the US
National Science Foundation (Division of Materials
Research) over many years (including the current grant
DMR-9903321) that made a sustained effort in the area
of elevated temperature crystalline plasticity realizable.
Part of this manuscript was written when the author
was on a short sabbatical leave at the Paul Scherrer
Institute (PSI) and Ecole Polytechnique Federale Lau-
sanne (EPFL) in Switzerland. The kind hospitality of
Dr. Maximo Victoria and Dr. Helena Van Swygen-
hoven of PSI and Prof. Jean-Luc Martin of EPFL is
acknowledged.
The author would like to acknowledge with thanks
the very efficient assistance of Ms. Debbie Snyder with
preparation of manuscripts that are usually brought in
at the eleventh hour, including this one. He would like
to thank his wife for her understanding of the author’s
habit of hiding in his cluttered home study.
This manuscript is dedicated to the memory of the
author’s former colleague, the late Professor John Emil
Dorn—a caring teacher and a path-breaking re-
searcher—of the University of California, Berkeley
campus.
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