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Acta Materialia 57 (2009) 5730–5738

Viscous grain-boundary sliding with rotating particles or grains

Byung-Nam Kim a,*, Keijiro Hiraga a, Koji Morita a, Hidehiro Yoshida a, Byung-Wook Ahn b

a National Institute for Materials Science, 1-2-1 Sengen, Tsukuba, Ibaraki 305-0047, Japanb Hanbat National University, SAN16-1, DuckMyoung-Dong, Yuseong-Gu, Daejeon 305-719, South Korea

Received 2 April 2009; received in revised form 3 August 2009; accepted 3 August 2009Available online 25 September 2009

Abstract

We evaluate the sliding rate and stress distribution on the boundary when the sliding of grain boundary containing particles is accom-modated by grain-boundary diffusion, by taking the particle rotation and the intrinsic boundary viscosity into account. The particle rota-tion enhances the sliding rate, and can occur in the reverse direction with respect to the grain-boundary sliding. We investigate the slidingbehavior for various particle shapes and boundary viscosities. A similar analysis is also conducted for the shear deformation of regularhexagonal grains.� 2009 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

Keywords: Grain boundaries; Diffusion; Grain rotation; High-temperature deformation; Analytical methods

1. Introduction

When nanocrystalline or fine-grained materials arestressed at high temperatures, diffusional deformationmay occur through the cooperation of diffusion andgrain-boundary sliding. The strain misfit caused by grain-boundary sliding is often accommodated by grain-bound-ary/lattice diffusion in order to maintain microstructuralcoherency. Raj and Ashby [1] first analyzed the cooperativemotion for the sliding of a non-planar grain boundary.They described the boundary by using a Fourier series,and obtained an analytical solution for the sliding ratedepending on the boundary configuration. In the analysis,however, they neglected the intrinsic boundary viscosityand assigned the effective boundary viscosity to the non-planarity of the boundary. Later, Schneibel and Hazzledine[2] reanalyzed the sliding rate by using the proceduresdeveloped by Spingarn and Nix [3], which are based onthe stress analysis on the grain boundary discretized withline segments. In the analysis, however, they did not con-sider the intrinsic boundary viscosity and could not obtain

1359-6454/$36.00 � 2009 Acta Materialia Inc. Published by Elsevier Ltd. All

doi:10.1016/j.actamat.2009.08.001

* Corresponding author.E-mail address: kim.byung-nam@nims.go.jp (B.-N. Kim).

an analytical solution for the sliding of curved grainboundaries. The sliding of arbitrarily shaped grain bound-ary with intrinsic boundary viscosity was recently analyzedby the present authors [4]. We obtained the analytical slid-ing rate by applying the energy-balance method developedby Mori et al. [5,6]. Despite its simplicity, the method givesan exact solution to the diffusional deformation problemwhere viscous sliding of grain boundary occurs in a steadystate. Viscous sliding induces shear stresses on the grainboundary, and affects the sliding rate and stress distribu-tion there.

Raj and Ashby [1] also analyzed the sliding of a grainboundary containing second phase particles. With anelectrostatic analogy, they formulated the rate of the grain-boundary sliding accommodated by diffusion around theparticles. For this problem, Mori et al. [5] combined theblocking effect of elastically deforming particles and the dif-fusional accommodation around the particles, and Kim et al.[4] analyzed the sliding behavior for various particle shapesby taking the intrinsic boundary viscosity into account. Allthese analyses, however, assumed non-rotating particles.Shear stresses acting on the periphery of particles duringgrain-boundary sliding can induce particle rotation.Although grain rotation can occur during annealing due to

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Fig. 1. Sliding of grain boundary containing rhombic particles.

B.-N. Kim et al. / Acta Materialia 57 (2009) 5730–5738 5731

grain misorientation and elastic anisotropy of surroundinggrains [7,8], it seems that higher torque acts on particles/grains during grain-boundary sliding, and this could be thereason why grain rotation is frequently observed duringhigh-temperature deformation [9,10]. In superplastic defor-mation for large elongations, grain rotation is consideredto occur as a necessary accommodation process for continu-ous grain-boundary sliding.

Theoretically, grain rotation during deformation wasexplained by dislocation motion [11]. A gliding dislocationon the grain boundary during grain-boundary sliding cansplit into two climbing dislocations at a triple junction,and the continuous dislocation climb on two parallel grainboundaries accompanies lattice rotation in a grain. The dis-location mechanism for grain-boundary sliding and grainrotation, however, has only limited application in somecases. For example, grain-boundary sliding in ceramicswith a glassy boundary is believed to occur in a viscousmanner [12,13], and some materials at low applied stressesare deformed by diffusion rather than by dislocation[14,15]. Since diffusional deformation including grain-boundary sliding accommodated by diffusion occurs atany stress level, whether it is the rate-controlling mecha-nism or not, grain rotation induced by viscous boundarysliding can also occur at any stress level during high-tem-perature deformation.

In the present study, we analyze the sliding of the vis-cous grain boundary containing particles, around whichdiffusional accommodation occurs. We calculate the ratesof both particle rotation and grain-boundary sliding, andalso investigate the effect of the particle shape on the rates.From the analysis, we realize that the present grain-bound-ary sliding can result in particle rotation which is com-pletely different from that predicted by frictional sliding.Finally, we apply the method to the shear deformation oftwo-dimensional polycrystals composed of regular hexago-nal grains.

2. Sliding of grain boundary containing particles

We first analyze the sliding of the viscous grain bound-ary containing particles with various shapes. Analyticalsolutions are derived in a steady state for respective particleshapes and compared to each other. Described below is thedetailed procedure for solving the case of rhombic parti-cles, because of its simple configuration.

2.1. Rhombic particles

Let us consider the grain boundary on which the rhom-bic particles of a facet length l and an inclination angle /are placed with an interval of k, as shown in Fig. 1. Themacroscopic shear stress s applied to the grain boundarycauses a relative sliding rate _U between the two adjoininggrains and a rotation rate _h of the particles in a steadystate. From the particle configuration shown in Fig. 1,the distance r from the rotation center of the particle to

the particle surface is ro/cos h, where h is the angle aroundthe rotation center and ro is the distance between theboundary facet and the rotation center (=l sin / cos /).Then, the distance s from the center of the boundary facetis given by

s ¼ ro tan hþ ð1=2� sin2 /Þl; ð1Þas a function of h. Since the particle rotation induces therelative velocity of the boundary facet, the normal and tan-gential components (vn,rot and vs,rot) of the relative velocitydue to the particle rotation are obtained as

vn;rot ¼ r _h1

rdrdh

� �1þ 1

rdrdh

� �2" #�1=2

¼ s _h� lo_h ð2Þ

and

vs;rot ¼ r _h 1þ 1

rdrdh

� �2" #�1=2

¼ �ro_h; ð3Þ

where lo is (1/2 � sin2 /)l. The relative velocity due to theparticle rotation is superimposed on that of the overallgrain-boundary sliding [4]. We then obtain the normaland tangential components (vn and vs) of the relative veloc-ity of the boundary facet as

vn ¼ �_U2

sin /þ s _h� lo_h ð4Þ

and

vs ¼_U2

cos /� ro_h; ð5Þ

on boundary facet of the particle. From the symmetry ofthe particle shape, we consider only boundary facet . Onthe grain boundary with a length of k–L between the par-ticles, vn = 0 and vs = _U , where L is the width of the parti-cle (=2lcos/). When lattice diffusion is ignored, the normalcomponent vn of the relative velocity is accommodated bygrain-boundary diffusion and the tangential component vs

by viscous grain-boundary sliding.For grain-boundary diffusion, the volume conservation

of the diffused matter demands vn = �XdJ/ds, where X isthe atomic volume and J is the flux of matter, namely thenumber of atoms per unit time per unit thickness in thedirection normal to the plane of the figure. If the thick-ness of the grain boundary is d, then J is related to thevelocity of diffusing atoms va as XJ = dva, so that the

5732 B.-N. Kim et al. / Acta Materialia 57 (2009) 5730–5738

normal velocity of the grain boundary vn can be rewrittenas vn = �ddva/ds [16]. On the other hand, since diffusionis driven by the gradient of the normal stress rn (or thechemical potential), the conservation equation that governsthe normal velocity vn can also be represented as vn = XdD/(kT) (d2rn/ds2), where D is the diffusion coefficient of theboundary, k is Boltzmann’s constant and T is the absolutetemperature. Hence, integrating vn = �ddva/ds andvn = XdD/(kT) (d2rn/ds2) with Eq. (4) under the boundarycondition of rn = 0 at s = ± l/2, we obtain

dva ¼ �_h2

s2 þ_U2

sin /þ lo_h

� �sþ l2 _h

24ð6Þ

and

XdDrn

kT¼ �

_h6

s3 þ 1

2

_U2

sin /þ lo_h

� �s2 þ l2 _h

24s

� l2

8

_U2

sin /þ lo_h

� �: ð7Þ

On the grain boundary between the particles, va and rn

are zero.According to the Einstein’s relationship, the velocity va is

related to the force F acting to the atom as va = FD/(kT).Since the work done by the diffusing atom, or, equivalently,the energy-dissipation rate _q of the atom, is given by_q = Fva, the energy-dissipation rate _W D due to diffusionalong the grain boundary is

_W D ¼ 4dX

Z l=2

�l=2

_qds ¼ 4dkTXD

Z l=2

�l=2

v2ads ð8Þ

for the four boundary facets of the rhombic particle.The above dissipation is caused by the normal stress

which alters the chemical potential of the atoms and drivesdiffusion. There is corresponding energy dissipation causedby the shear stress which drives grain-boundary sliding.This term, the energy-dissipation rate _W S due to viscousboundary sliding, is _W S =

Rssvsds, where ss is the shear

stress on the boundary. If viscous sliding is described bya Newtonian flow, ss is represented as gvs/d, where g isthe intrinsic boundary viscosity. Here, we assume that theviscosity g is the same for all the boundaries. Then _W S bothin the four boundary facets of the particle and in the grainboundary with a length of k–L between the particles isobtained as

_W S ¼ 4gd

l_U2

cos /� ro_h

� �2

þ gd

_U 2ðk� 2l cos /Þ: ð9Þ

The total energy _W dissipated by the overall grain-bound-ary sliding shown in Fig. 1 is the sum of _W D of Eq. (8) and_W S of Eq. (9), and is dependent on the rotation rate _h of the

particle. Since _W is a quadratic function of _h, the particlerotation ð _hÞ can reduce the energy required for the grain-boundary sliding and occurs in order for _W to be a mini-mum. The rotation rate _h of the particle to yield a minimum_W can therefore be obtained from d _W =d _h ¼ 0, and is

represented as

_h ¼ H_Ul; ð10Þ

where

H ¼ �30ð1=2� sin2 /Þ sin /þ 360 sin / cos2 / � g�

1þ 60ð1=2� sin2 /Þ2 þ 720 sin2 / cos2 / � g�; ð11Þ

and g* is the normalized grain-boundary viscosity definedas gXD/(kTl2). We use the normalized viscosity g* in orderto evaluate the relative contribution of the intrinsic bound-ary viscosity g to the overall grain-boundary sliding. In thepresent model, the viscous boundary sliding and the grain-boundary diffusion are independent processes that act inparallel with the slower one controlling the overall slidingrate. The overall sliding rate for the parallel action is rep-resented as a combination of the two components, whereg appears in the term gXD/(kTl2) and D appears in theother term without g. Therefore, g* represents the relativeintensity of the intrinsic boundary viscosity with respectto the diffusional contribution. g* can also be representedas g/gD l/d, where gD, defined as kTl3/(XdD), is the effectiveviscosity for diffusional deformation.

The rotation rate _h of the particle can also be obtainedfrom a condition of moment equilibrium. Since themoment M of the particle represented as

M ¼ 4

Z l=2

�l=2

rnðlo � sÞdsþ 4sslro ð12Þ

should be zero in a steady state, we obtain results identicalto Eqs. (10) and (11) by solving Eq. (12) at M = 0. Therotation rate _h is dependent on the particle shape and theboundary viscosity. In Eqs. (10) and (11), it is noteworthythat the value of _h can take a negative value under certainconditions, indicating reverse rotation against the slidingdirection of grain boundary. The details of the reverse rota-tion are discussed later.

The total energy dissipated by both grain-boundary diffu-sion and viscous grain-boundary sliding should be suppliedby the external work in a steady state. For the overall slidingof the grain boundary containing the rhombic particlesshown in Fig. 1, the energy balance therefore gives

ks _U ¼ _W D þ _W S: ð13ÞUsing Eqs. (8)–(10) to solve _U in Eq. (13), we obtain

_U ¼ GXdDs

kTl2¼ G _U l; ð14Þ

where _U l is the normalized sliding rate defined as XdDs/(kTl2) and

G¼ kl

1

180þ1

3

1

2� sin2 /

� �2" #

H 2þ1

3

1

2� sin2 /

� �sin/H

(

þ 1

12sin2 /þg� ð4sin2 /H 2�4sin/H þ1Þcos2 /

�þ k

l�2cos/

� ����1

: ð15Þ

B.-N. Kim et al. / Acta Materialia 57 (2009) 5730–5738 5733

The rate constant G in Eq. (15) is inversely proportional tog*, so that the sliding rate _U in Eq. (14) decreases monot-onously with increasing boundary viscosity g, as in the caseof non-rotating particles [4]. However, the effect of the par-ticle shape (l and /) on _U is quite complicated because ofthe strong dependence of H on the particle shape, as shownin Eq. (11).

Fig. 2 represents the rate constant Gð¼ _U= _UlÞ, which var-ies with the particle shape h/L for constant L/k(= 0.5), whereh is the height of the particle. For comparison, we representthe cases of rotating and non-rotating (H = 0) particles inFig. 2a and b, respectively. From the two figures, we can eas-ily see that the sliding rate of the grain boundary is enhancedby the particle rotation. To understand the variation in thesliding rate G in Fig. 2, consider the case of constant L andk, where an increasing h means an increase in the particle sizeand /. For non-resistant boundary sliding (g* = 0), G

decreases with increasing h as a whole, due to an increasein the diffusion distance along the boundary facet. The com-plicated variation around h/L = 1 in Fig. 2a, which is relatedto the particle rotation, is discussed later. For g* = 1, the slid-ing rate G increases slightly with increasing h at h/L < 1.47,

10-1

100

101

102

103

104

105

0.001 0.01 0.1 1 10 100 1000

η*=0

0.1

0.01

1

Nor

mal

ized

slid

ing

rate

, G

Particle shape, h/L

(a)

Nor

mal

ized

slid

ing

rate

, G0

Particle shape, h/L

Fig. 2. Variation of the normalized sliding rate G (G0) with respect to theparticle shape h/L at L/k = 0.5: (a) for rotating and (b) non-rotating(H = 0) particles.

then decreases at h/L > 1.47. The low G at low h/L is dueto the increased energy dissipation by viscous boundary slid-ing. For particles elongated in parallel with the sliding direc-tion (h/L� 1), the overall grain-boundary sliding is mainlycontrolled by the boundary viscosity, whereas it is mainlycontrolled by the diffusion for vertically elongated particles(h/L� 1). Since an increase in h (or /) reduces the energydissipation required for viscous sliding (Eq. (9)), for g* = 1,G increases with h at h/L < 1.47. A further increase in h over1.47L, however, requires high energy dissipation for diffu-sion, which results in lowering the sliding rate G, as shownin Fig. 2.

We now consider the particle rotation during the grain-boundary sliding. From Eqs. (10) and (14), the rotationrate _h of the rhombic particles for constant macroscopicshear stress s can be rewritten as

_h ¼ HG_U l

l: ð16Þ

Fig. 3a represents the normalized rotation rateHGð¼ _hl= _U lÞ varying with the particle shape h/L at L/k = 0.5. For most particle shapes, the rotation rate HG

-100

-50

0

50

100

150

200

250

0.001 0.01 0.1 1 10 100 1000

0.01

1

L/λ=0.5

0.1

Nor

mal

ized

rota

tion

rate

, HG

Particle shape, h/L

η*=0

0.03

(a)

Particle shape, h/L

Nor

mal

ized

rota

tion

rate

, H

Fig. 3. Variation of the normalized rotation rate: (a) HG at a constantmacroscopic shear stress s(L/k = 0.5) and (b) H at a constant sliding rate_U (independent of k).

(a) (b)

Fig. 4. Schematic normal stresses and particle rotation for negligible shearstresses (g = 0). The torque induced by the normal stresses rotates theparticle: (a) in the reverse direction and (b) in the sliding direction,depending on the particle shape. and indicate the tensile and thecompressive stresses, respectively.

0

0.5

1

1.5

2

0 0.02 0.04 0.06 0.08 0.1

sliding direction

reverse direction

Parti

cle

shap

e, h

/L

Boundary viscosity, η*

Fig. 5. Critical particle shape for the transition of the rotating direction.

5734 B.-N. Kim et al. / Acta Materialia 57 (2009) 5730–5738

decreases with increasing boundary viscosity g*. Consider-ing that the viscous boundary sliding induces shear stresseson grain boundary and contributes to particle rotation, wecan expect an increasing rotation rate with increasingboundary viscosity. We must consider, however, that thesliding rate G decreases with the boundary viscosity forconstant s, as shown in Fig. 2. Since the effect of theboundary viscosity on G is larger than that on H, the largerthe boundary viscosity, the lower the rotation rate HG formost rhombic particles. An exception is found around h/L = 1 in Fig. 3a, where the g*-dependence of the rotationrate HG is quite complicated. At the two extreme particleshapes, h/L = 0 and h/L =1, the particle rotation stops(HG = 0) regardless of the value of g*.

For a constant sliding rate _U , on the other hand, therotation rate of the particles is represented by Eqs. (10)and (11). From Eq. (11), the normalized rotation rateHð¼ _hl= _UÞ is independent of the particle interval k, whichis different from the above case for constant s. The normal-ized rotation rate H varying with the particle shape h/L isshown in Fig. 3b. The rotation rate H approaches 0 forparticles elongated in parallel with the sliding direction(h/L� 1) and approaches 0.94 for vertically elongated par-ticles (h/L� 1). At h/L = 1.07, H represents a constantvalue of 0.68, independently of the boundary viscosity g*.

An interesting point in Fig. 3 is that, under certain con-ditions, the particle rotates in the reverse direction againstthe sliding direction of grain boundary, which had not pre-viously been expected. Since viscous sliding induces shearstresses on grain boundary as frictional sliding, the particlerotation is usually expected to occur in the direction favor-able to the sliding. However, when diffusional deformationis accompanied, we have to consider the distribution of thenormal stress on the grain boundary, which also affects thetorque acting to the particle. As an example for the direc-tion of particle rotation, let us consider the simple case ofg* = 0 and non-rotating particles. Owing to the free slidingof boundary (g* = 0), there exist only normal stresses onthe grain boundary: compressive stresses on boundaryfacet and tensile stresses on boundary facet in Fig. 1.Both the compressive and tensile stresses on the boundaryfacets induce torque in the particle. At h/L > 1, the net tor-que acts in the sliding direction, whereas at h/L < 1, it actsin the reverse direction. For an equiaxed particle (h/L = 1),the torque vanishes. Hence, for particles elongated in par-allel with the sliding direction (h/L� 1) and low g*, thereverse rotation of particles can occur. With an increasein the aspect ratio of the particle, the net torque varies con-tinuously and the direction changes from counterclockwiseto clockwise at h/L = 1. Fig. 4 schematically represents therotating direction depending on the particle shape forg* = 0. In more general cases, whether the particles rotatein the sliding or reverse direction is dependent on the par-ticle shape, size, boundary viscosity, diffusional coefficientand temperature, because g*, defined as gXD/(kTl2) or g/gD l/d, represents the relative contribution of the intrinsicboundary viscosity.

The critical particle shape (h/L) at which the transitionof the rotating direction occurs depends on the value ofg*. The value of h/L for the transition decreases withincreasing g*, and disappears at g* � 0.042 (Fig. 3b). Forg* > 0.042, the particle rotates in the sliding direction atany h/L-values for rhombic particles because of theincreased shear stress due to viscous boundary sliding.The critical particle shape (h/L) for the transition withrespect to the boundary viscosity (g*) is shown in Fig. 5.The reverse rotation occurs only for low boundary viscos-ities and for particles elongated in parallel with the slidingdirection. Here, it should also be noted that g* (=gXD/(kTl2)) is inversely proportional to l2. Not only low g butalso large l lowers the value of g*. This means that thereverse rotation can occur for large rhombic particles dur-ing conventional grain-boundary sliding accommodated bygrain-boundary diffusion. Although the reverse rotationhas not been confirmed experimentally, the present analysispredicts the possibility theoretically.

Using the characteristics of the particle rotation shownin Fig. 3, we can understand the complicated behavior ofthe sliding rate G around h/L = 1 in Fig. 2a. As mentionedabove, the particle rotation stops at h/L = 1 for g* = 0 dueto the vanishing torque, so that the sliding rate G is equal

B.-N. Kim et al. / Acta Materialia 57 (2009) 5730–5738 5735

to the value for non-rotating particles (G0). Since the rota-tion occurs at the other particle shapes (h/L – 1) to reducethe energy required for the overall grain-boundary sliding,the sliding rate G for rotating particles increases with adeviation from the equiaxed shape of h/L = 1, whichappears as a hollow around h/L = 1 for g* = 0 in Fig. 2a.Where g* > 0.042 such complicated behavior of G doesnot appear, because the rotation occurs in the sliding direc-tion for all particle shapes. Furthermore, the rotation rateis relatively low at high boundary viscosities, as shown inFig. 3a, so that the difference between the sliding ratesfor rotating and non-rotating particles decreases.

The normal stress on the boundary facet of rotating par-ticles during grain-boundary sliding is shown in Fig. 6 forthe case of L/k = 0.5 and h/L = 0.5. With increasingboundary viscosity, the maximum normal stress decreasesalong with the sliding rate (Fig. 2). As found from Eq.(7), the normal stress is a cubic function of s for rotatingparticles, whereas it is a quadratic function for non-rotat-ing particles. For this reason, compressive and tensile stres-ses can exist simultaneously on the same boundary facet forrotating particles. The tensile stress which appears onboundary facet for g* = 0 is due to the reverse rotationof the particle. Though the normal stress for non-rotatingparticles is compressive on boundary facet , the reverserotation partly induces the tensile stress on the boundaryfacet far from the rotation center. For particle rotation inthe sliding direction (g* > 0.042), the normal stress onboundary facet becomes compressive on the entire facet.

2.2. Other particles

We applied the present energy-balance method to thesliding of the viscous grain boundary containing rectangu-lar, hexagonal or circular particles. For the case of rectan-gular particles, the grain-boundary sliding and particlerotation showed behavior similar to those of rhombic par-ticles (Figs. 2 and 3). That is, the complicated behavior of

Fig. 6. Distribution of the normal stress rn on the boundary facet at h/L = 0.5 and L/k = 0.5.

the sliding rate appears around the equiaxed particle shape,and the transition of the rotation from the reverse to thesliding direction occurs with an increase in the ratio ofthe particle height to the particle width for low boundaryviscosities.

Fig. 7 plots the sliding rates of the grain boundary con-taining particles with various shapes: equiaxed rhombic,equiaxed rectangular (square), hexagonal and circular par-ticles. The particles have the same cross-sectional area A

and period k. For comparison, the normalized sliding rate_UA and the boundary viscosity g�A have been defined asXdDs/(kTA) and gXD/(kTA), respectively. For g�A ¼ 0,the equiaxed rhombic, hexagonal and circular particlesdo not rotate during grain-boundary sliding. Only thesquare particle rotates due to the non-vanishing torqueeven for g�A ¼ 0. Fig. 7 compares the effect of the particleshape on the rate of grain-boundary sliding for the cover-age ratio

pA/k = 0.5.

The sliding rate _U= _UA decreases with an increase in theboundary viscosity g�A. At low boundary viscosities, thesliding rate is in order square > circular > hexago-nal > rhombic particles, whereas the order is reversed athigh boundary viscosities. For particles with a height-to-width ratio of 1 (square, circular and rhombic particles),the sliding rate is inversely proportional to the particleheight at low boundary viscosities. At low boundary vis-cosities, the grain-boundary sliding is controlled by diffu-sion, the distance of which is proportional to the particleheight. Hence, the sliding rate of grain boundary for lowheight particles becomes higher than that for high heightparticles: in Fig. 7, the sliding rate at g�A < 0:01 is in ordersquare > circular > rhombic particles, where the height/p

A is 1, 2/p

p andp

2, respectively. At high boundary vis-cosities, however, the grain-boundary sliding is controlledby the boundary viscosity. The particles with larger widthhave higher torque, higher rotation rate and hence highersliding rate, so that the sliding rate at g�A > 0:1 is reversedto be in order rhombic > circular > square particles.

Fig. 7. Comparison of the sliding rates of the grain boundary containingparticles with various shapes for

pA/k = 0.5.

5736 B.-N. Kim et al. / Acta Materialia 57 (2009) 5730–5738

3. Shear deformation of hexagonal grains

Finally, we analyzed the shear deformation of a poly-crystal composed of regular hexagonal grains, as shownin Fig. 8. For shear deformation of the hexagonal grains,there are two sliding directions (modes I and II), whichare shown by the thick lines in the figure. Paidar andTakeuchi [17] had analyzed the shear deformation formode II only, when grain-boundary sliding occurs by dislo-cation motion and grain rotation occurs geometrically orcompulsorily in order for grain boundaries to keep perpen-dicular between the centers of two adjoining grains. In thepresent study, we conducted an analysis for both modesusing the above energy-balance method, when grain-boundary sliding occurs in a viscous manner and grainrotation occurs by the net torque acting to the grains.

During the shear deformation of the polycrystal, all thehexagonal grains rotate in the sliding direction at the samerate. Due to the simultaneous rotation of adjoining grainsin the same direction, the contribution of the grain rotationto the normal component (vn) of the relative velocity van-ishes, and as a result the normal stress on the grain bound-ary becomes a quadratic function of s, not a cubic function.For the same reason, the contribution of the grain rotationto the tangential component (vs) of the relative velocity istwice that for the case of a single rotation.

Solving the problem with the boundary condition that anet flux of diffused matter is zero at triple junctions, weobtain the sliding rate _U as

_U ¼ XdDs

kTL2

72

1þ 36g�Lðmode IÞ ð17Þ

and

_U ¼ XdDs

kTL2

72ffiffiffi3p

1þ 36g�Lðmode IIÞ; ð18Þ

where L is the grain size and g�L ¼ gXD=ðkTL2Þ. The slidingrate for mode II is

p3 times higher than that for mode I.

This is because the amplitude of the sliding boundary islower for mode II, as also discussed in Fig. 7. The amplitude

Fig. 8. Shear deformation of hexagonal grains.

of the sliding boundary is L/2p

3 for mode II, whereas it isL/2 for mode I. The interval of the sliding planes, however,is shorter for mode I (L/2) than for mode II (

p3L/2).

Hence, we obtain the same shear strain rate _c for the twosliding modes as

_c ¼ XdDs

kTL3

144

1þ 36g�L: ð19Þ

For g�L ¼ 0, the shear strain rate is essentially identicalto the results of Raj and Ashby [1]. In their results, onlythe numerical value of 144 in Eq. (19) is replaced by 132,along with g�L ¼ 0. In the analysis of Raj and Ashby [1],it should be noted that the grains are non-rotating, and amathematical approximation, described in detail elsewhere[4], was used to solve the problem. For non-rotating (fixed)hexagonal grains, on the other hand, only the shear compo-nent (vs) of the relative velocity is affected and the viscosityterm 36g�L in Eqs. (17)–(19) is replaced by 108g�L. Due to thelarger energy dissipation required for the shear deforma-tion of the non-rotating grains, the sliding and shear strainrates become lower than those for the rotating grains.

The distribution of the normal stress on the grainboundary is also identical for the two modes. For each slid-ing mode, there are three kinds of grain boundary with dif-ferent normal stresses: compressive, tensile and zerostresses, shown as , and in Fig. 8, respectively. Calcu-lating the normal stress rn on the compressive boundary ,we obtain as

rn

s¼ s

l

� 2

� 1

4

�6ffiffiffi3p

1þ 36g�Lð20Þ

for both sliding modes, where l is the length of the bound-ary facet (=L/

p3) and s is the distance from the center of

the boundary facet (�l/2 6 s 6 l/2). On the tensile bound-ary , the sign of Eq. (20) is reversed. Fig. 9 shows the dis-tribution of the normal stress on the grain boundary formode II. With the boundary viscosity ðg�LÞ, the normalstress rn decreases due to decreasing sliding rate _U . Themaximum tensile stress for g�L ¼ 0 is rn = 1.5

p3s.

-3

-2

-1

0

1

2

3

Nor

mal

stre

ss, σ

n/τ

LocationQOP

P QO

ηL*=0

1

0.01

0.1

Fig. 9. Distribution of the normal stress rn during mode II sliding ofhexagonal grains.

B.-N. Kim et al. / Acta Materialia 57 (2009) 5730–5738 5737

The grain rotation rate _h is also identical for both slidingmodes:

_h ¼ XdDs

kTL3

72

1þ 36g�Lðg�L – 0Þ: ð21Þ

From Eqs. (19) and (21), the grain rotation rate is 1/2times the shear strain rate ( _h = _c/2) and is independent ofthe boundary viscosity except for g�L ¼ 0. Compared withthe geometrical relationship ( _h = 3_c/4) [17], the presentrotation rate is lower by 2/3 times. For g�L ¼ 0, the net tor-que vanishes and grain rotation stops due to the symmetricgrain shape in the present model, whereas grain rotationstill occurs in the geometrical model irrespective of theboundary viscosity [17]. It is also noteworthy that _h is pro-portional to L�3 during the shear deformation accommo-dated by grain-boundary diffusion. When grain rotationoccurs by the reduction of anisotropic boundary energyduring annealing and is accommodated by grain-boundarydiffusion, _h varies as L�4 [4]. During actual deformation athigh temperatures, both grain rotations would occursimultaneously.

From the rotation rate _h = _c/2 of the hexagonal grainsduring shear deformation, we can estimate the grain rota-tion during tensile deformation of the hexagonal grains.Let us assume that the tensile deformation occurs only byshear in one direction under maximum shear stress (45�withrespect to the tensile axis). Within the sample, there arenumerous sliding boundaries in this direction, so that thesample is uniformly deformed at a strain rate of _e (= _c/2).We also assume that the microstructure is invariant dur-ing the deformation. The rotation angle then becomes thesame with the tensile strain (h = e). Apparently, the esti-mated rotation angle must be an upper limit. For actualpolycrystals, the effective torque of a three-dimensionalgrain would be lower, and the grain-boundary slidingwould occur in several directions for the net rotation rateto be reduced significantly. Nevertheless, this estimationis useful in predicting the maximum rotation angle duringsuperplastic or creep deformation. Vevec�ka and Langdon[18] measured the grain rotation during superplastic defor-mation of a Pb–62%Sn alloy, and reported a maximumrotation of �20� at an elongation of 100%. This grainrotation is about half that of the present estimation fortwo-dimensional hexagonal grains. Considering the three-dimensional grain shape of the alloy and the grain-bound-ary sliding in several directions during deformation, thepresent analysis provides a reasonable estimation.

In the shear deformation of regular hexagonal grains,the grain rotation occurs in the sliding direction, andreverse rotation does not occur. In a polycrystal, reverserotation can occur for large grains elongated in parallelwith the sliding direction, as in the case of the rhombic par-ticles elongated along the sliding boundary (Fig. 4a). The-oretically, reverse rotation can contribute to the alignmentof elongated grains during uniaxial deformation. Forexample, consider the elongated grains inclined withrespect to the tensile axis. The elongated grain would

provide the preferable sliding direction along the long axis,because of high sliding rates in that direction. For low g�L,reverse torque can act on the elongated grains, resulting inreverse rotation. The elongated grains would then becomealigned in parallel with the tensile axis.

Grain rotation that is favorable to the sliding direction,on the other hand, can suppress grain elongation and main-tain the equiaxed grain shapes when diffusional deforma-tion of grains occurs simultaneously. It is well knownthat grains remain the equiaxed shapes during superplasticdeformation even after huge elongations exceeding 1000%,and the mechanism has been explained by grain switching[19]. The favorable grain rotation accompanied by diffu-sional deformation, however, can explain the equiaxedshapes during superplastic deformation. For example, con-sider again the elongated grains inclined with respect to thetensile axis. In contrast to the reverse rotation, the favor-able rotation makes the elongated grains perpendicular tothe tensile axis, and then the concurrent diffusional defor-mation of the grains reduces the aspect ratio [16,19].Hence, by the concurrent processes of grain rotation anddiffusional deformation, the grains can keep their equiaxedshapes during superplastic deformation.

4. Concluding remarks

The sliding behavior of viscous grain boundary contain-ing particles with various shapes was analyzed by takingthe particle rotation and the intrinsic boundary viscosityinto account. The particle rotation reduces the energyrequired for grain-boundary sliding and enhances the slid-ing rate, whereas the boundary viscosity increases theenergy for viscous sliding and reduces the sliding rate.Although the boundary viscosity causes shear stressesand induces torque in the particle, the effect of the torqueis lower than the effect of the reducing sliding rate. As aresult, the increasing boundary viscosity reduces the rota-tion rate of the particles. The present analysis also predictsthe reverse rotation of particles with respect to grain-boundary sliding. The particles elongated in parallel withthe sliding direction can rotate in the reverse direction forlow boundary viscosities.

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