Materials Science and Engineering A 409 (2005) 125–130

Grain growth and deformation in nanocrystalline materials

Chandra S. Pande∗, Robert A. MasumuraNaval Research Laboratory, Code 6325, Washington, DC 20375, USA

Received in revised form 13 April 2005; accepted 16 April 2005

Abstract

In case of nanocrystalline materials several additional features need to be taken into account in considering grain growth and deformation whoseexistence was first anticipated by Professor J.C.M. Li years ago. Grains can increase their size by grain rotation as well as by curvature drivenmotion. Grain rotation has been considered in detail by Li [J.C.M. Li, J. Appl. Phys. 15 (1962) 2958] theoretically and recently other researchersby simulation. We show that in nanocrystalline materials this mode is quite possible especially if the grain growth is retarded by finite triplejunction mobility. New deformation modes occur in these materials leading to the so-called inverse Hall–Petch effect, which will be discussed indetail.© 2005 Elsevier B.V. All rights reserved.

K

1

gwaiaIbaulfbadnifav

aboved for

ent

the

h a

niteain,

120ame

0d

eywords: Grain growth; Hall–Petch; Nanomaterials

. Introduction

This paper briefly reviews our work in the area of grainrowth and deformation of nanomaterials. Grain growth is theell known phenomenon of the evolution of microstructure indeformed polycrystal after recrystallization resulting in the

ncrease in average grain size, by the motion of grain bound-ries due to annealing at a certain temperature and time[1–5].

n particular in classical grain growth, the importance of grainoundary area/length reduction with time, motion by curvaturend the role of dihedral angles is now considered central to thisnderstanding. In two-dimensional grain growth, these concepts

ead to well-known von Neumann law[6] initially formulatedor bubble growth whose validity for grain growth was provedy Mullins [7]. In nanomaterials, these concepts may not bepplicable in toto and may need modification. Similarly, in theeformation studies of these materials, one observes behaviorot seen in large grain materials, and thus one needs to mod-

fy our view of the deformation process since it may be differentrom that for large grained polycrystal. In both these areas, viewsnd concepts introduced long back by Prof. J.C.M. Li provide

2. von Neumann–Mullins law

If An(t) is the area of the grain withn sides at any instantt, andM is a constant, then under the assumptions mentionedand further discussed later, the following relation is deducetwo-dimensional grain growth[6,7]:

dAn(t)

dt= M(n − 6). (1)

This relation has played an important role in the developmof the theory of grain growth.

In obtaining von Neumann–Mullins law, Mullins usedfollowing four basic assumptions of grain growth process:

(a) All the grain boundaries during grain growth move witvelocity proportional to its curvature.

(b) The triple junctions of the grain boundaries have infimobility during any change in number of sides of the gretc.

(c) The boundaries meet at grain boundaries at an angle of.(d) All the grain boundaries are isotropic and have the s

aluable guidelines.

∗ Corresponding author. Tel.: +1 202 767 2744; fax: +1 202 767 2623.

grain boundary energies.

It should be realized that all the four assumptions are built inthe von Neumann–Mullins law of grain growth and so the useof this law is equivalent to accepting these assumptions. If thel t von

E-mail address: pande@anvil.nrl.navy.mil (C.S. Pande).

921-5093/$ – see front matter © 2005 Elsevier B.V. All rights reserved.oi:10.1016/j.msea.2005.04.058

ast two conditions are relaxed, it can be easily shown tha

126 C.S. Pande, R.A. Masumura / Materials Science and Engineering A 409 (2005) 125–130

Neumann–Mullins law will not hold exactly for each grain butit may in a statistical sense.

In case of nanocrystalline materials two additional factorsneed to be taken into account:

(a) grain rotation (increases grain growth);(b) finite triple junction mobility (retards grain growth).

These two mechanisms will operate in addition to curvaturedriven grain growth. Grain rotation has been considered in detailby Li [8] theoretically and by Haslam et al.[9] by simulation. Wewill not consider this any further except to state that in nanocrys-talline materials this mode is quite possible if the grain growthis retarded by triple junction finite mobility. How grain growthcould be retarded will be considered in detail below.

2.1. Triple junction mobility as function of the number ofsides

Gottstein and Shvindlerman[10] have shown that a parame-ter,λ defined as

λ = (grain size) (triple junction mobility)

grain boundary mobility(2)

which involves the ratio of the mobilites of the triple junction tot

λ

f

w eaI illa sl

Fig. 2. Growth rate forn-sided grain.

2.2. Modified von Neumann–Mullins law

When the triple junction mobility is infinite,λ is infinite andthe triple junction angle, 2θ, is equal to 120. However, whenλis finite, the triple junction can deviate from 120. Thus, the vonNeumann–Mullins law requires modification. Using the aboveresults, we plot, dAn/dt as a function ofλ for various values ofn inFig. 2(λ is defined by Eq.(2)). It is seen that for 1.43≤ λ ≤ 10.73,dAn/dt is positive for alln; thus, the grains will tend to grow,which is not possible, since the total area of the polycrystal (fora two-dimensional system) is conserved. Grain growth is onlypossible if some grains decrease in size and disappear. Therefor sufficiently smallλ grain growth will be retarded. Sinceλinvolves the grain size, the smaller grain sizes will have relativelysmallerλ. Thus, in nanocrystalline materials grain growth couldbe retarded. It should be remembered that the present discussionis strictly applicable to two dimensions (“thin film”) only. Graingrowth in two dimensions do share common features with threedimensions, so some of the results may be applicable in threedimensions also, but that remains to be proved. Experimentalevidence for grain retardation has been provided by Okuda et al.[11] for thin films.

From this analysis, it appears that modified von Neumann–Mullins approach to account for a finite triple junction mobil-ity can result in grain growth retardation in nanocrytallinematerials.

3

er

τ

w indi-v ast qua-t eter.

he grain boundary and can be expressed as

= ln[sin(θ)]

2 cos(θ) − 1for n > 6 and λ = 2θ

2 cos(θ) − 1

or n < 6, (3)

here ln[·] is the natural logarithm, 2θ the triple junction anglndn is the number os sides. This relation is illustrated inFig. 1.

t is seen that if the mobility is finite, the triple junction angle wlways be different from 120 and the von Neumann–Mullin

aw is only approximately obeyed.

Fig. 1. Triple junction angle dependence onλ.

. Deformation of nanocrystalline materials

The classic Hall–Petch relationship[12,13] describes thelationship between yield stressτ and grain sized, viz.

= τ0 + kd−1/2 (4)

hereτ0 is the friction stress considered needed to moveidual dislocations andk is a constant (often referred tohe Hall–Petch slope and is material dependent). This eion is well behaved for grains larger that about a microm

C.S. Pande, R.A. Masumura / Materials Science and Engineering A 409 (2005) 125–130 127

Masumura et al.[14] have plotted some of the available datain a Hall–Petch plot. It is seen that the yield stress–grain sizeexponent for relatively large grains appears to be very close to−0.5 and generally this trend continues until the very fine grainregime (∼100 nm) is reached. With the advent of nanocrystallinematerials whose grain sizes are of nanometer (nm) dimensions,the applicability and validity of Eq.(1) becomes of interest inview of recently compiled experimental results[15].

A close analysis of experimental Hall–Petch data in a varietyof materials shows that although the plot ofτ versusd−1/2 formsa continuous curve, three different regions can be seen, viz.: (1)a region from single crystal to a grain size of about a micrometer(m) where the classical Hall–Petch description can be used; (2)a region for grain sizes ranging from about a 1m to about 30 nmwhere the Hall–Petch relation roughly holds, but deviates fromthe classical−0.5 exponent to a value near zero; (3) a regionbeyond a very small critical grain size where the Hall–Petchslope is nearly zero with no increase in strength on decreasinggrain size or where the strength actually decreases with decreas-ing grain size. Although some of the measurements on whichthe trend discussed above is based on are not entirely reliablebecause of a variety of reasons discussed recently by Sanders etal. [15]. The above delineation into three regions is beginning tobe accepted. This paper is mostly interested in the mechanismapplicable to the third (lowest grain sizes) region. However, it ispossible to obtain expression for yield stress applicable to anyg

3

nismb rains nbI nd-a rioum tsa conc oringg ypeo islo-c ionst etchti hera pile-u akena odei mals n-t r theHw hichi ystau ptioi dt ode

is still applicable with the sole exception that the analysis musttake into account of the fact that in the nanometer size grains thenumber of dislocations within a grain cannot be very large.

If the number of dislocations,n, in a pile-up is not too large,the length of the pile-upL is not linear inn but an additionalterm is necessary[25], Pande and Masumura[24] using a resultfrom Szego[26] give an improved expression, viz.

L = A

2τ

[2(n + m − 1)1/2 − i1 + ε

(12)1/3(n + m − 1)1/6

]2

(5)

whereA = Gb/(πτ* ), G is the shear modulus,b the Burgers’vector, τ* the barrier stress,mb the Burgers vector of thefirst dislocation behind which other dislocations are piling up,i1/61/3 = 1.85575 andε is a small correction term (ε 1) and canbe neglected. For small grain sizes, this gives additional termsto Hall–Petch relation[13]. This model however cannot explaina drop inτ for very fine-grained materials. In what follows, forsake of simplicity we ignore this correction.

One can of course assume that dislocation sources must oper-ate in each grain, and so an additional component of the yieldstress exists of at leastGb/d. However, as shown by Yamakov etal. [27] such a possibility is not likely. There are other disloca-tion model such as due to Valiev et al.[28], Malygin [29] andGryaznov et al.[30]. The latter proposed a generalization of theof Hall–Petch relationship. By making judicious assumptionsa tiont

eptt nismo e fors s int bero they ablyh

ofc ch ana hichht them videa stressf dO rainb

τ

w ouldo

3

tchm t pos-s lated

rain sizes.

.1. Dislocation models

For large grain sizes most of the models use a mechaased on dislocations. They account very well for the gize dependence of the stress,τ in Eq. (4); most of these cae rationalized in terms of a dislocation pile-up model[16].

n deriving the Hall–Petch relation, the role of grain bouries as a barrier to dislocation model is considered in vaodels. In one type of model[17–19], the grain boundary acs a barrier to pile-up of dislocations, causing stresses toentrate and activating dislocation sources in the neighbrains, thus initiating slip from grain to grain. In the other tf models[20,21] the grain boundaries are regarded as dations barriers limiting the mean free path of the dislocathereby increasing strain hardening, resulting in a Hall–Pype relation (for a survey, see Lasalmonie and Strudel[22]). Its clear that a variety of mechanisms could be postulated. Tre several problems with any pile-up model. Existence ofps in all types of models, both bcc and fcc cannot be ts established. Even in the framework of a conventional m

t can be shown that it cannot be applicable to grains of sizes[23]. Pande and Masumura[24] by considering the conveional Hall–Petch model showed that a dislocation theory foall–Petch effect gives a linear dependence ofτ on d−1/2 onlyhen there are large number of dislocations in a pile-up, w

s equivalent to assuming that the grain sizes in the polycrnder consideration is large. What happens if this assum

s not strictly true? Pande and Masumura[24] have investigatehe case where classical Hall–Petch dislocation pile-up m

s

-

,

e

ll

ln

l

nd approximations, they were able to develop a formulahat can account for yield stress for any grain sizes.

We will not discuss dislocation models any further exco state that at still smaller grain sizes, dislocation mechaperating in the interior of the grain should cease becausufficiently small grain size there will be only two dislocationhe pile-up. And eventually for even finer grain sizes the numf dislocations will falls to one and no further increase inield stress is possible and it should saturate. This probappens at grain sizes less than 15 nm[14].

Lack of dislocation activity in the interior of the grainsourse does not preclude activity in the grain boundaries. Suctivity can lead to some very interesting consequences, wave been investigated by Gutkin and Ovid’ko[31]. They find

hat such dislocation activity is very much dependent onode of preparation of the specimens. If true, this could pron explanation of the large scatter in the data seen in yield

or smaller grain sizes as seen in[14]. Specifically, Gutkin anvid’ko show that the contribution to yield stress due to goundary dislocations is given by,

= k1 + k2

d(6)

herek1 andk2 are constants. This expression of course shnly be applicable for very fine grains.

.2. Mechanism involving Coble creep

Clearly, at sufficiently small grain sizes, the Hall–Peodel based upon dislocations may not be operative excep

ibly in the grain boundaries. It has however been postu

128 C.S. Pande, R.A. Masumura / Materials Science and Engineering A 409 (2005) 125–130

that in this range of grain sizes, a new mechanism of deforma-tion may be operative akin to the familiar Coble creep or grainboundary diffusional creep, however in this case acting at evenroom temperatures[14,32]. Coble creep is a deformation pro-cess that leads to homogeneous elongation of grains along thetensile direction. It is believed to be strain rate dependent andwhere the strain rate is usually given as,

εC = cτΩδDDgb

kBTd3 (7)

whereδD is the width of the diffusing channel (approximatelyequal to the grain boundary width),Dgb the diffusion constantfor a grain boundary,T the temperature,kB the Boltzmann’sconstant,τ the applied stress,Ω the activation volume andc isa proportionality constant that depends upon the grain shape.From Eq.(6),

τ = Bd3, where B = εkBT

cΩδDDgb. (8)

Chokshi et al.[33] first proposed room temperature Coble creepas the mechanism to explain the so-called inverse Hall Petcheffect seen in nanocrystalline specimens. Certainly, there is anorder of magnitude agreement and the trend is correct, however,the functional dependence ofτ on d found by them is incorrectas pointed out by Neih and Wadsworth[32]. Conventional Coblec 3 −1/2 −6

τ isi ar

τ

w lyt

pro-v dt sizeo rains l thua atura re if andN andbb

ε

w

ε

wia enm( of

the Nabarro–Herring creep to the yield stress will be small ascompared to Coble creep and can be neglected in most cases.

The statistical nature of the grain sizes in a polycrystal is takeninto consideration by using an analysis similar to Kurzydlowski[34]. The volume of the grains are assumed to be log-normallydistributed. Finally, it is assumed that a grain sized* exists atwhich value of grain size the classical Hall–Petch mechanismswitches to the Coble creep mechanism, i.e.,τhp = τc at d = d* .This model gives an analytical expression forτ as a function ofthe inverse square root ofd in a simple and approximate mannerthat could be compared with experimental data over a wholerange of grain sizes.

A major consideration in this approach is what explicitexpression to use for Coble creep. Eq.(7) was not found to besuitable since it led to an extremely steep drop ofτ with d−1/2.In the model of Masumura et al.[14], theτ versusd relationshipused for Coble creep is given by

τc = B0

d+ Bd3 (12)

whereB0 is a constant, whileB is both temperature and strain-rate dependent. This threshold term[35] B0/d can be large ifd is in the nanometer range. For intermediate grain sizes, bothmechanisms (dislocations and Coble creep) might be active ifthe specimen has grain size distribution, as is usually the case.

We have also considered the expression developed byY sso e notp creepp ckingw en-t eC ives as ts okshiea

4

formo

τ

w eralp firstt r thata et rei

τ

S willb ant,t dary

reep demands as shown above thatτ ∼ d = [d ] , i.e., theversusd−1/2 curve falls very steeply asd−1/2 increases. Th

s not found experimentally[33]. Their data however fit betterelation of the form

= β − K′d−1/2 (9)

hereβ andK′ are constants. Eq.(9) cannot be related simpo any known mechanism.

A plausible explanation for this experimental fact wasided by Masumura et al.[14]. In their model, it is assumehat that polycrystals with a relatively large average grainbey the classical Hall–Petch relation. For very small gizes, it is assumed that Coble creep is active. The modessumes that diffusion is significant even at room tempernd contributes to the yield stress through creep. There a

act two different mechanisms of creep, viz. Coble creepabarro–Herring creep corresponding to grain boundaryulk diffusion, respectively. The Coble creep rate (Eq.(7)) cane re-written as

˙C ∼(

Dgb

d3

)τΩ

kT, (10)

hile Nabarro–Herring formulation results in a creep rate,

˙NH ∼(

Dbulk

d2

)σΩ

kT, (11)

hereDgb is the grain boundary diffusion coefficient andDbulks the corresponding coefficient in bulk. SinceDgb Dbulk by

factor of about 106, εC εNH. The difference becomes evore pronounced for smaller grains because of thed3 in Eq.

10) versus thed2 in Eq. (11). Therefore, the contribution

sen

amakov et al.[27] for Coble creep for calculating yield stref nanocrystalline materials. The experimental results arrecise enough to compare the two expressions for Coblerecisely. We have compared the two expressions by chehich of the two versions are consistent with the experim

al finding of Chokshi et al.[33] for copper and palladium. Thoble creep expression developed with a threshold stress gomewhat better fit unless the values ofd* are relatively small. Ihould however be noted that the experimental result of Cht al. [33] is some what controversial (see references[35,36])nd hence further experimental results are needed.

. A generalized expression for yield stress

Collecting various terms discussed above, a generalizedf yield stress applicable to any polycrystal is obtained as

= τ0 + kd−1/2 + k1 + B0

d+ Bd3, (13)

herek1 and B0 are as yet undetermined constants. Sevoints regarding this equation should be noted. Firstly, the

wo terms on the left-hand side is to be used for grains largegiven critical sized* . For grains smaller thand* , the last thre

erms on the right-hand side should be applicable. As befod*

s obtained by the following equation:

0 + k(d∗)−1/2 = k1 + B0

d∗ + B(d∗)3 (14)

econdly, not all the three terms on the right-hand sidee significant for every situation. If Coble creep is domin

he last two terms will be dominant, whereas if grain boun

C.S. Pande, R.A. Masumura / Materials Science and Engineering A 409 (2005) 125–130 129

dislocations are active, the first two terms may be more signifi-cant. Eq.(13)differs from the corresponding result used in[14]in two respects. First, it has an additional termk1 and second, theB0/d term, which was introduced in an ad hoc fashion in[14],is identified with grain boundary dislocation reactions. Finally,this equation is true provided all the grains are of the same size.In a real polycrystal, we will have a distribution of grain sizes.Hence, the averaging procedure given by Masumura et al.[3]should be used.

The technique applicable to the present situation is brieflydescribed below. We define a volume average of the system ofEq.(13) following Kurzydlowski[34] as

(τ)ave− τ0 = 1

mv

∫ ∞

d∗kd−1/2vf (v) dv + 1

mv

∫ d∗

0

×(

k1 + B0

d+ Bd3

)vf (v) dv, (15)

wherev is the grain volume andf (v) is the grain volume dis-tribution function and is assumed to be lognormal. The grainvolume averagemv is defined by

mv =∫ ∞

0vf (v) dv ≈ γd3, (16)

whered is the mean grain size andγ is a geometrical factor oftio iveni

e anat l, wp

(

wn be

e

I

w alc eforba

I

w tM rve

of (τ)ave versusd−1/2 should peak aroundd* . This givesM1 as

M1 = K

6(d∗)5/2

. (20)

Thus, finally

(τ)ave = τ0 + K

(d)1/2 − K

6(d∗)1/2

(d∗

d

)3

× exp

[−M2

ln

(d

d∗

)2]

. (21)

We propose this approximate, semi-empirical equation as a con-venient and simple equation to describe the whole range of grainsizes. The first two terms, of course, give the classical Hall–Petchresult. The last term is responsible for the Inverse Hall–Petcheffect.M2 is about 1,K (the Hall–Petch slope) can be obtainedexperimentally andd* being given by Eq.(15).

It is interesting to note that in this scheme, the mechanismsresponsible for the Inverse Hall–Petch effect can, as a firstapproximation, be represented byd* alone. In principled* canbe obtained in terms ofd from the peak,dp, of the experimental

curve ofτ versus (d)−1/2. If the experimental gain size distri-bution is very narrow,dp ≈ d* anddp or d* is given by Eq.(15).If however the grain size distribution is not narrow, the actualp −1/2 *

u gingp td

τ

w

τ

he order of one. All the integrals in Eqs.(15) and(16) can bentegrated exactly in terms of error functions[14]. A comparisonf the prediction with experiments for copper and nickel is g

n [14], where it is shown that the model works well.For practical purposes, it might be convenient to hav

pproximate expression but simpler than developed in[14] forhe yield stress as a function of grain size. Towards that goaroceed as follows. Eq.(15)can be re-written as

τ)ave− τ0 = 1

mv

∫ ∞

0kd−1/2vf (v) dv − 1

mv

∫ d∗

0

×(

kd−1/2 − k1 − B0

d− Bd3

)× vf (v) dv or (τ)ave− τ0 = I1 − I2, (17)

hereI1 is the first integral andI2 is the second integral.When using a lognormal distribution, the first integral ca

valuated and is equal to

1 = k exp(−5σ2/72)

(d)−1/2 = K

(d)−1/2 , (18)

hereσ is the standard deviation in ln(v). The second integran be expressed in terms of error functions as mentioned but we find that for the range of grain sizes from 0 tod* , a goodpproximation is given by,

2 = M1

(d)3exp

[−M2

ln

(d

d∗

)2]

, (19)

hereM1 andM2 strictly are functions ofd*. But we find tha2 ≈ 1 andM1 can be obtained from using the fact that the cu

e

e,

eakdp of τ versus (d) curve may not coincide withd . Bysing various grain size distributions and using the averarocedure given by Masumura et al.[14], it is easy to show thap ≥ d* . In this case, Eq.(18) is only approximately true.

Taking these factors into account, we write Eq.(21)as

norm = ξ − M1ξ6 exp[−4M2ln(ξ)2], (22)

here

norm = (τ)ave− τ0

K(dp)−1/2 and ξ =(

d

dp

)−1/2

, (23)

Fig. 3. Simplified expression for yield stress (Eq.(17)).

130 C.S. Pande, R.A. Masumura / Materials Science and Engineering A 409 (2005) 125–130

and because of the reasons stated before,M1 ≈ 1/6 andM2 ≈ 1.In Fig. 3, we plot Eq.(22)using these values ofM1 andM2. Theright side of the curve will be affected if somewhat differentvalues ofM1 andM2 are used.

It is seen that this figure that Eq.(22)is able to account for boththe conventional and “inverse Hall–Petch” regions. We shouldpoint out that Eq.(22) is semi-empirical, and is being presentedhere as a rough approximation to the actual function obtainedby the detailed averaging procedure[14] involving complicatederror functions. Eq.(22) uses just two constants,M1 and M2whose values are approximately given and uses the normaliza-tion of the average grain size with the peak of the curve, whichis easily obtained from the experimental data.

5. Summary and conclusions

We have investigated theoretically, the connection betweengrain size and yield stress (or microhardness) for polycrystals,especially of nanometer sizes. We obtain an expression for theyield stress as a function of grain size, applicable to any materialand for any grain size. Depending on the mode of the prepara-tion, not all the terms in this equation will contribute. Hence fornanocrystalline materials, a large scatter in a Hall–Petch plotis expected, as is experimentally seen. We also specify, whichof the terms will dominate for a given mode of preparation ofthe nanocrystalline material. Finally, a semi-empirical equationr ed.

R

leve-

[7] W.W. Mullins, J. Appl. Phys. 27 (1956) 900.[8] J.C.M. Li, Appl. Phys. 33 (1962) 2958.[9] A.J. Haslam, S.R. Phillpot, D. Wolf, Mater. Sci. Eng. A 318 (2001) 293.

[10] G. Gottstein, L.S. Shvindlerman, Scripta Mater. 38 (1998) 1541.[11] S. Okuda, M. Kobiyama, T. Inami, S. Takamura, Scripta Mater. 44

(2001) 2009.[12] E.O. Hall, Proc. Phys. Soc. Lond. B64 (1951) 747.[13] N.J. Petch, J. Iron Steel Inst. 174 (1953) 25.[14] R.A. Masumura, P.M. Hazzledine, C.S. Pande, Acta Mater. 46 (1998)

4527.[15] P.G. Sanders, J.A. Eastman, J.R. Weertman, Acta Mater. 45 (1997)

4019.[16] J.C.M. Li, Y.T. Chou, Met. Trans. 1 (1970) 1145.[17] T.E. Mitchell, S.S. Hecker, R.L. Smialek, Phys. Stat. Sol. 11 (1965) 585.[18] R.W. Armstrong, A.K. Head, Acta Met. 13 (1965) 759.[19] C.S. Pande, R.A. Masumura, Proceedings of the Sixth International Con-

ference on Fracture, 1984, p. 857.[20] N. Louat, Acta Met. 33 (1985) 59.[21] A.G. Evans, J.P. Hirth, Scripta Met. 26 (1985) 1675.[22] A. Lasalmonie, J.L. Strudel, J. Mater. Sci. 21 (1986) 1837.[23] C.S. Pande, R.A. Masumura, R.W. Armstrong, Nanostruct. Mater. 2

(1993) 323.[24] C.S. Pande, R.A. Masumura, in: C. Suryanarayana, J. Singh, F.H. Froes

(Eds.), Processing and Properties of Nanocrystalline Materials, TMS,Warrendale, PA, 1996, p. 387.

[25] Y.T. Chou, J. Appl. Phys. 38 (1967) 2080.[26] G. Szego, Orthogonal Polynomials, vol. 23, American Mathematical

Society Colloquium Publications, 1939.[27] V. Yamakov, D. Wolf, S.R. Phillpot, H. Glieter, Acta Mater. 50 (2002)

61.[28] Z. Valiev, E.V. Kozolv, Yu.F. Inanov, J. Lian, A.A. Nazarov, B. Bran-

[[ Sci.

[[[ 89)

[[[ B 60

elating yield stress with average grain size is also propos

eferences

[1] H.V. Atkinson, Acta Metall. 36 (1988) 469.[2] S.K. Kurtz, F.M.A. Carpay, J. Appl. Phys. 51 (1980) 5725.[3] C.S. Smith, Trans. AIME 175 (1948) 15.[4] C.S. Smith, Trans. AIME 194 (1949) 755.[5] C.S. Smith, Acta Metall. 1 (1953) 295.[6] J. von Neumann, Metal Interfaces, American Society for Metals, C

land, OH, 1952, p. 108.

delet, Acta Mater. 42 (1994) 2467.29] G.A. Malygin, Phys. Solid State 37 (1995) 1248.30] V.G. Gryaznov, M.Yu. Gutkin, A.E. Romanov, L.I. Trusov, J. Mater.

28 (1993) 4359.31] M.Yu. Gutkin, I.A. Ovid’ko, private communication, 2002.32] T.G. Nieh, J. Wadsworth, Scripta Met. 25 (1991) 955.33] A.H. Chokshi, A. Rosen, J. Karch, H. Gleiter, Scripta Met. 23 (19

1679.34] K.J. Kurzydlowski, Scripta Met. 24 (1990) 879.35] S.M.L. Sastry, private communication, 1997.36] H.V. Swygenhoven, M. Spaczer, A. Caro, D. Farkas, Phys. Rev.

(1999) 22.