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Conductivity of disordered polycrystalsPham Duc Chinh Citation: Journal of Applied Physics 80, 2253 (1996); doi: 10.1063/1.363053 View online: http://dx.doi.org/10.1063/1.363053 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/80/4?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Resistanceexpansiontemperature behavior of a disordered conductor–insulator composite Appl. Phys. Lett. 69, 2602 (1996); 10.1063/1.117713 On the conductance and the conductivity of disordered quantum wires J. Appl. Phys. 80, 3876 (1996); 10.1063/1.363343 A microstructural study of carbon fibers for thermal management and space applications AIP Conf. Proc. 361, 869 (1996); 10.1063/1.49979 Critical current and texture relationships in YBa2Cu3O7 thin films AIP Conf. Proc. 165, 132 (1988); 10.1063/1.37113 Transport properties of boron carbide AIP Conf. Proc. 140, 206 (1986); 10.1063/1.35596
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Conductivity of disordered polycrystalsPham Duc ChinhInstitute of Mechanics, 224 Doi Can, Hanoi, Vietnam
~Received 20 February 1996; accepted for publication 8 April 1996!
New upper and lower bounds are constructed for the macroscopic conductivity of polycrystals withrandom microstructure, given the principal conductivities of the constituent crystals~and the volumefractions of phases in case of a multiphase polycrystal!. The new bounds lie inside the well-knownHashin–Shtrikman ones. ©1996 American Institute of Physics.@S0021-8979~96!03914-X#
I. INTRODUCTION
Voight1 and Reuss,2 in their respective classical works,suggested the arithmetic and harmonic means of the princi-pal conductivities of the basic crystal as approximations forthe macroscopic conductivity of the isotropic polycrystallineaggregate formed from it. Following Hill3 in using the re-spective classical variational principles, one can show thatVoigt’s and Reuss’ average values are the upper and lowerbounds of the conductivity of polycrystals, provided that theconstituent crystals are distributed equally in all directions~this requirement, though being natural, in fact is more thanthe simple assumption of macroscopic isotropy in conductiv-ity. Generally, the bounds are valid for all macroscopicallyisotropic polycrystals!. New lower bound for the conductiv-ity of isotropic polycrystals has been established4 and shownto be optimal,5 while Schulgasser6 found a simple model thatattains the upper arithmetic mean bound. The cell con-structed by Schulgasser is formed from a few laminate crys-tals bonded in a prescribed arrangement. It possesses isotro-pic conductivity and, if we view a polycrystal as a singlecell, the crystals there are not distributed equally in all direc-tions. One can form a polycrystal with no preference in crys-tal orientation by randomly combining the cells of Schulgas-ser. We call such a configuration a locally orderedpolycrystal. Though such polycrystals theoretically could bemanufactured, it is likely that in most practical cases of poly-crystalline formation the crystals are formed independentlyand meet each other at random~in the sense of their relativecrystalline and shape orientations!. Conductivity of supposedto be real disordered polycrystals is the subject of this study.Naturally they also possess overall isotropic conductivity,however the class of disordered polycrystals should be muchmore restrictive than the larger class of macroscopically iso-tropic polycrystals. For such real disordered polycrystals,Hashin and Shtrikman7 established the bounds, which aretighter than the upper arithmetic bound, and—over a largeinterval of parameters—the lower Avellanedaet al. bound4
~see Ref. 8!. In the following, we explore further the disorderhypotheses to derive new bounds on the macroscopic con-ductivity of disordered polycrystals, which are closer thanthe Hashin–Shtrikman ones. The bounds for general aniso-tropic composites composed of anisotropic components andthose with probabilistic features of the microstructure havebeen constructed elsewhere.4,9,10
II. CONSTRUCTION OF THE BOUNDS
Consider a representative element of a multiphase poly-crystal that occupies spherical regionV of Euclidean three-dimensional spaceR3. The center of the sphereV is also theorigin of the Cartesian system of coordinates$x1 ,x2 ,x3%.The representative element consists ofn components occu-pying regionsVa,V-each component is composed of crys-tals of the same phase and crystal directions, the conductivityof which is described by the second-order tensorsa ~a51,...,n!. A polycrystal-uniphase or multiphase could be rep-resented by suchn-component configuration with any degreeof approximation whenn→`. Let na denote the volume ofVa and, for convenience, it will be assumed that the volumeof V is unit. The polycrystalline aggregate is supposed to bemacroscopically isotropic and its effective conductivitysc
can be defined as follows:
sce0•e05 inf
^e&5e0I ~e!, I ~e!5E
Ve•s•edx, ~1!
wheree~x! is a vector field~e.g., electric field, temperaturegradient,...!, which is the gradient of a continuous functionon V; e0 is any average vector
e05^e&5EVedV; s~x!5(
asaka~x!;
ka~x! is the characteristic function taking the value 1 inVa
and 0 inR3\Va ; the Greek letters under the sign of sum runon natural numbers from 1 ton.
In Eq. ~1! we take the condition of fixed average of thefield. It can be shown that the infimum point is homogeneousin currents on the boundary ofV and satisfies equilibriumequations inV.
Our objective is to find a best admissible fielde so thatwith the help of Eq.~1! we can deduce a best possible upperbound onsc .
We can rewrite the functional~1! in the following form:
I ~e!5EVe•s0
•edx1(a
va^e&asa0•^e&a1U~e!, ~2!
where
U~e!5(a
EVa
~e2^e&a!•sa0•~e2^e&a!dx, ~3!
^e&a51
vaEVa
edx, s05s0E, sa05sa2s0,
2253J. Appl. Phys. 80 (4), 15 August 1996 0021-8979/96/80(4)/2253/7/$10.00 © 1996 American Institute of Physics [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP:
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s0 is any positive constant,E is the unit second-order tensor.Take a transformation of the second term in Eq.~2! by in-troducing variablespa
(a
va^e&a•sa0•^e&a52(
ava^e&a•p
a2F~p!,
pa5sa0•^e&a , ~4!
F~p!5(a
vapa•~sa0!21
•pa.
Now instead of functional~1! we shall consider a new func-tional
I p~e!5EV~e•s0
•e12p•e!dx2F~p!1U~e!, ~5!
in the following class of polarization fields:
p5(a
paka , pa5const.
From the above formulas one can see
I ~e!5I p~e! ~6!
whenp satisfies Eq.~4!. Take a change of variables
e5e01e8, p85(a
p8aka , p8a5pa2^p&,
then Eq.~5! has the form
I p~e!5JV~e8!1e0•s0•e012^p&^e&12e0•s0
•^e8&
2F~p!1U~e!, ~7!
where
JV~e8!5EV~e8•s0
•e812p8•e8!dx.
We take for an admissible fielde85e8—the minimum pointof the functional
J`~e8!5ER3
~e8•s0•e812p8•e8!dx,
which should satisfy equation~remember thate8 is the gra-dient of a function!:
~s0ei81pi8!, i50, ei850S 1
uxu2D when uxu→`,
whereei8 andpj8a are the components of vectorse8 andp8a;
summation is carried on repeating Latin indices from 1 to 3;Latin indices after comma denote differentiation with respectto corresponding coordinates. One finds
e i8521
s0(a
pj8aw ,i j
a , ~8!
where the functionswa denote the harmonic potentials
wa~x!521
4p EVa
ux2yu21dy, ¹2wa5ka . ~9!
Thus, we have
^e&5^e81e0&5e0,
J`~ e8!521
3s0(a
vap8a•p8a
52(a
vap8a•~s0* !21
•p8a, ~10!
s*5s*E, s*52s0 , s0*5s01s* . ~11!
To obtain Eqs.~10! and ~11! we admitted the following hy-pothesis for disordered polycrystals:
^w ,i ja &b5
1
vbEVb
w ,i ja dx5 1
3dabd i j , ~12!
wheredab and di j are the usual Kronecker delta. Equation~12! is called a directional-disorder condition. It comes froman assumption of isotropic distribution of all the componentsin V ~see Refs. 11 and 12!.
Since 2p8•e8 vanishes outsideV, one can see thatJV~e8!<J`~e8!. Substitutinge5e5e01e8 into Eq.~5! and no-ticing ^e8&50, one gets
I P~ e!<J`~ e8!1e0•s0•e012^p&•e02F~p!1U~ e!
5F2(a
vap8a•~s0* !21•p8a1e0•s0
•e012^p&•e0
2(a
vapa•~sa0!21
•paG1U~ e!.
Let pa5pa be the stationary point of the quadratic form insquare brackets in the last inequality
pa5@E2s0* •~sa* !21#•F(b
vb~sb* !21G21
•e0,
p8a5HE2~sa* !21•F(
bvb~sb* !21G21J •s0* •e0, ~13!
sa*5sa1s* .
One can verify thatpa and^e&a satisfy Eq.~4!, therefore Eq.~6! holds. Note that e&5e0—so e is an admissible field ofEq. ~1!. Thus
sce0•e0<I ~ e!
5I p~ e!<e0•H F(a
va~sa* !21G21
2s* J •e01U~ e!.
~14!
Similar to Eq. ~12!, the directional-disorder hypothesis re-quires that~take into account Eq.~9!; consult also Ref. 13!
1
2EVa
@~w ,i jb 2^w ,i j
b &a!~w ,klg 2^w ,kl
g &a!1~w ,i jg 2^w ,i j
g &a!
•~w ,klb 2^w ,kl
b &a!#dx5Aabg 1
10~d ikd j l1d i ld jk223d i jdkl!,
~15!
where
2254 J. Appl. Phys., Vol. 80, No. 4, 15 August 1996 Pham Duc Chinh [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP:
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Aabg5E
Va
@~w ,i jb 2^w ,i j
b &a!~w ,i jg 2^w ,i j
g &a!dx. ~16!
U~e! in Eqs.~14! and ~3! is evaluated with the help of Eqs.~8! and ~15!
U~ e!51
s02 • (
a,b,gs i j
a0pk8bpl8
gEVa
~w ,ikb 2^w ,ik
b &a!
3~w , j lg 2^w , j l
g &a!dx
51
10s02 • (
a,b,gAa
bg~s i ia0pj8
bpj8g1 1
3s i ja0pi8
bpj8g!. ~17!
Now we try to evaluate geometric parametersAabg from Eqs.
~16! and~9! for our disordered polycrystal. Ann-componentcomposite is called structurally disordered if there exists adivision of the composite into portions of equal volumes andeach portion is composed of the material of only one com-ponent, such that the interchange of materials of any twodifferent portions would not affect the composite macro-scopically. Particularly, each componentVais divided intopa portions Vak (k51,...,pa) of equal volume v0(va5pa•v0) and our structural-disorder hypothesis requiresthat
EVak
w ,i jakw ,i j
akdx2v03
5 f 1
5const for all ak511,...,1p1 ,...,npn ,
EVak
w ,i jakw ,i j
b l dx5 f 2 for all akÓb l ~ i.e.,aÞb or kÞ l !,
EVak
w ,i jb lw ,i j
b l dx5 f 3 for all akÓb l ,
EVak
w ,i jb lw ,i j
gmdx5 f 4 for all akÓb lÓgmÓak.
Where
wak~x!521
4p•E
Vak
ux2yu21dy.
~Remember that conventional summation is carried only onrepeating Latin lower indicesi . j !.
It is natural to expect that if the composite is disorderedwith a division $pa% of Eq. ~18!-type then there would bedivisions$r •pa% also of Eq.~18!-type, wherer is any naturalnumber, sopa can be chosen as proportionally great asneeded. Our definition of a structurally disordered materialhas the same sense as those of Miller’s symmetric cellmaterial14 and Bruno’s infinitely interchangeable material.15
Detailed analysis of disordered multicomponent materialscomposed of isotropic components is given in Ref. 16, wherethe bounds for such materials are obtained, which reduce toMiller’s ones in the two-component case.
Denote
f 15f 1v0, f 25
f 2v02 , f 35
f 3v02 , f 45
f 4v03 .
It is clear thatf i do not depend on the volume fractionsva :we could change the material of any portion ofVak , and bythat change the volume fractionsva , but this does not affectf i from Eq. ~18!.
Now we could evaluate the geometric parametersAabg
from Eqs.~16! and ~9! @aÞbÞgÞa#
Aabg5E
Va
w ,i jb w ,i j
g dx
5papbpg•EVa1
w ,i jb1w ,i j
g1dx5papbpg• f 4
5vavbvg•f 4v03 5vavbvg• f 4 ,
Aaab5E
Va
w ,i ja w ,i j
b dx5papbEVa1
w ,i ja w ,i j
b1dx
5papb•F EVa1
w ,i ja1w ,i j
b1dx1~pa21!•EVa1
w ,i ja2w ,i j
b1dxG5papb•@ f 21~pa21! f 4#
5vavb•F f 2v02 1va•pa21
pa•
f 4v03G5vavb•~ f 21va f 4!,
Aabb5vavb•~ f 31vb f 4!,
~19!Aa
aa5va•~ f 112va f 21va f 31va2 f 4!.
In the above equalities, we take 1 instead of (pa21)/pa
because, as already noted, we could choosepa as proportion-ally great as we need.
Our next step is to find the relations amongf i . From thetheory of harmonic potential one has~remember thatV is asphere!
w~x!521
4p•E
Vux2yu21dy5
x•x
61const, xPV.
Thus,
051
3•E
Va
w ,i ia dx2
va
3
5EVa
w ,i jw ,i ja dx2
va
35(
bAa
ba
5va•@ f 11 f 21va~ f 21 f 31 f 4!#.
As f i are independent ofva , from the last equality one de-duces
f 252 f 1 , f 45 f 12 f 3 .
From Eq.~19! we obtained~aÞbÞgÞa!
2255J. Appl. Phys., Vol. 80, No. 4, 15 August 1996 Pham Duc Chinh [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP:
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Aabg5vavbvg~ f 12 f 3!,
Aaaa5va~12va!@~12va! f 11va f 3#,
~20!
Aaab5vavb@~va21! f 12va f 3#,
Aabb5vavb@~12vb! f 31vb f 1#.
BecauseAabb.0, one hasf 1.0 and f 3.0. Now substituting
Eq. ~20! into Eq. ~17! and taking into account
(b
vbpi8b5^pi8&50,
one derives
U~ e!51
10s02 •F(
ava~s i i
a0pj8apj8
a1 13s i j
a0pi8apj8
a! f 1
1S (a
vas i ia0•(
bvbpj8
bpj8b1 1
3(a
vas i ja0
•(b
vbpi8bpj8
bD f 3G , ~21!
wherep8a is defined in Eq.~13! and f 1 , f 3 are positive.The lower bound onsc , the counterpart of the upper
bound Eqs.~14! and ~21! is derived similarly from the dualvariational problem.
III. UNIPHASE POLYCRYSTAL
Then-component material considered would really rep-resent an uniphase disordered polycrystal when we letn→`(va→0) and the crystals to be distributed equally inall directions. Lets1,s2,s3 denote the principal conductivi-ties of the basic crystal. The conductivity tensor of a compo-nent with any crystal orientationsa can be represented as
sa5Ra•sd
•~Ra!T,
whereRa is an orthogonal matrix,~Ra!T its transpose,sd adiagonal matrix tensor
sd5diag$s1 ,s2 ,s3%.
We have
sa05sa2s05Ra•sd0
•~Ra!T, sd05sd2s0,
sa*5sa1s*5Ra•sd* •~Ra!T, sd*5sd1s* .
Because the crystals are distributed randomly inV, for anyscalar functionS~s! one has
(a
vaRa•diag$S~s1!,S~s2!,S~s3!%•~R
a!T
5^S~s!&E;
here and throughout this section, the average value of a sca-lar s-functionS~s! means particularly
^S~s!&5 13@S~s1!1S~s2!1S~s3!#. ~22!
Thus
F(a
va~sa* !21G21
2s*
5F(a
vaRa•~sd* !21
•~Ra!TG21
2s*5P~s* !E, ~23!
where
P~s* !5^~s1s* !21&212s* . ~24!
Taking the differentiation ofP~s*! with respect tos
*, we
have
dP~s* !
ds*5^~s1s* !21&22^~s1s* !22&21>0
⇔^~s1s* !22&>^~s1s* !21&2.
The last inequality is well known from mathematical analy-sis, in addition the ‘‘equality’’ takes place only in the trivialcase s15s25s3 ~remember that for our problems1,s2,s3,s*
.0!. SoP~s*! is an increasing function ofs
*.
p8a from Eq. ~13! can be rewritten as
p8a5~s01s* !Ra•@E2~P~s* !1s* !~sd* !21#
•~Ra!T•e0.
Now one can calculateU~e! from Eq. ~21! readily
U~ e!5S 310 f 1Qu~s0!1^s2s0&(b
vbpi8bpi8
b
•
1
3s02 f 3D e0–e0, ~25!
where
Qu~s0!5^~s2s0!@12^~s12s0!21&21~s12s0!
21#2&
19^s2s0&^@12^~s12s0!21&21~s12s0!
21#2&.
~26!
Finally, if one takess5s0u such that
^s&<s0u and Qu~s0
u!<0, ~27!
then from Eq.~25! one hasU(e)<0 and with the help ofEqs.~14! and~23! one deduces the following upper bound onsc :
sc<P~2s0u!. ~28!
The best available upper bound will be Eq.~28! with
s0u5min$s0u^s&<s0<smax,Q
u~s0!<0%, ~29!
where
smax5max$s1 ,s2 ,s3%, smin5min$s1 ,s2 ,s3%.
Looking at Eq.~26! we see thatQu~smax!,0. Because of thecontinuity ofQu~s0! on s0, we gets0
u,smax and, as a con-sequence, the new upper bound is less than the Hashin–Shtrikman bound
P~2s0u!,P~2smax!.
2256 J. Appl. Phys., Vol. 80, No. 4, 15 August 1996 Pham Duc Chinh [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP:
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~the conditions0<smax, which is not absolutely necessary,is included in Eq.~29! only to indicate thats0
u exists and liesinside the defined interval!.
Similarly, from the dual variational problem, one derivesthe lower bound onsc
sc>P~2s0l !, ~30!
where
s0l 5max$s0usmin<s0<^s21&21, Ql~s0!<0%, ~31!
Ql~s0!5 K S 1s21
s0D F12 K S 1s 1
1
2s0D 21L 21
3S 1s 11
2s0D 21G2L 19K 1s2
1
s0L
3 K F12 K S 1s 11
2s0D 21L 21S 1s 1
1
2s0D 21G2L .
~32!
Similarly, one can verify that the new lower bound is greaterthan the Hashin and Shtrikman bound
P~2s0l !.P~2smin!.
Looking at Eqs.~26!, ~24!, and ~29! and supposing that theconductivitiess1,s2,s3 of a crystal do not differ greatlyfrom each other, one can see that^s& is near tos0
u andP~2^s&! is very near toP(2s0
u). Similarly ^s21& is near to~s0
l !21 andP~2^s21&21! is very near toP(2s0l ). So we sug-
gest the following simple approximate bounds for practicaluses:
P~2^s21&21!<sc<P~2^s&!. ~33!
From Eqs.~28! and ~29! we see that the approximate upperbound in Eq.~33! would become exact whenQu~^s&!<0.Similarly from Eqs.~30! and ~31!, the approximate lowerbound in Eq. ~33! would become exact whenQl~^s21&21!<0. Additional support for the practical value ofEq. ~33! can be found in Ref. 17. For an illustrative examplewe takes151, s252, s353. Calculations yield:~1! The upper arithmetic mean bound~^s&! and the lowerbound of Avellanedaet al.4 ~the solutionss of the equation^(s2ss)/(2s1ss)&50!, which are valid for all macro-scopically isotropic polycrystals
1.759<sc<2.000
~2! Hashin–Shtrikman bounds for disordered polycrystals
1.830<sc<1.916
~3! Exploring the equiaxity of the polycrystals, Helsing18 ob-tained
1.836<sc<1.889
~4! The new bounds~28!, ~29!, ~30!, and~31!
1.871<sc<1.888
~5! The approximate bounds~33!
1.872<sc<1.888.
~s0l 51.611, near to^s21&2151.636,
P~2s0l !51.871, very near toP~2^s21&21!51.872).
We see that even thoughs1,s2,s3 differ considerably in ourexample, the approximate bounds are very near to the exactbounds~the upper bound is exact in this case!.
The new exact bounds~28!, ~29!, ~30!, and ~31! andapproximate bounds~33! have been derived on the assump-tion of disorder-hypothesis~12!, ~15!, and~18!. The hypoth-esis is natural for polycrystals with random distribution ofcrystals. One could imagine that the locally ordered poly-crystal of Schulgasser6 does not possess both the directional-disorder property~12!, ~15! and structural-disorder property~18!, e.g., each crystal chooses its neighbors in directionalpreference according to the structure of a local cell and if oneinterchanges the materials of any two components with twodifferent crystalline orientations and these two orientationshappen to be those of crystals from some Schulgasser cellsthen the structure of those cells would collapse and theywould no longer possess the prescribed optimal property.
Avellaneda and Bruno19 and Helsing8,18studied the classof equiaxed polycrystals, requiring only the symmetry in theinterchange of crystal axes. Mathematically the class of equi-axed polycrystals is larger than the class of disordered poly-crystals with random nature~about relative crystalline andshape orientations of the constituent crystals! consideredhere. There might exist certain equiaxed locally orderedpolycrystals not satisfying~12!, ~15!, or ~18!.
In the case of uniaxial base crystals, i.e, whens25s3,Eqs.~29! and ~31! are resolved explicitly
s0u5 H ~s112s2!/3
~7s1113s2!/20if s1<s25s3
if s1>s25s3, ~34!
s0l 5 H 3/~1/s112/s2!
20/~7/s1113/s2!
if s1>s25s3
if s1<s25s3. ~35!
The results of calculations of the bounds for various valuesof s1 ands25s3 are presented in the Table I. HeresHS
u andsHSl denote Hashin–Shtrikman7 bounds;sACLM
l , the lower
TABLE I. Comparison of the bounds on the macroscopic conductivity ofdisordered polycrystals formed from uniaxial base crystals withs2 5 s3 forvarious ratios ofs1 ands2. The new boundssu ands l refer to ~28!,~34!and ~30!,~35!; sHS
u and sHSl , denote Hashin–Shtrikmana bounds;
sHu ,sH
l —the bounds of Helsingb; sACLMl —the lower bound of Avellaneda
et al.
s1 s2,s3 sHSu sH
u su s l sHl sHS
l sACLMl
2 1 1.294 1.294 1.282 1.279 1.273 1.2365 1 2.073 2.067 1.913 1.777 1.706 1.53110 1 3.333 3.316 2.824 2.174 2.000 1.70820 1 5.838 5.797 4.564 2.499 2.213 1.832100 1 25.84 25.61 18.25 2.8790 2.435 1.9621 2 1.625 1.619 1.615 1.601 1.600 1.5621 5 3.378 3.299 3.120 2.898 2.846 2.7021 10 6.250 6.000 4.959 4.289 4.000 4.0001 20 11.97 11.35 7.281 6.104 5.071 5.8441 100 57.69 54.04 12.01 13.67 6.500 13.65
aReference 7.bReference 8.cReference 4.
2257J. Appl. Phys., Vol. 80, No. 4, 15 August 1996 Pham Duc Chinh [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP:
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Avellaneda et al.4 bound; sHu and sH
l , Helsing8 bounds~based on interpolation between the Hashin–Shtrikmanbounds and the Avellanedaet al. bounds!; su andsl are thenew bounds calculated from Eqs.~28!, ~34! and ~30!, ~35!.Note that the new bounds seem much tighter than all previ-ous bounds. However, at a large difference betweens1 ands2 ~e.g.,s151, s25100!, the lower bound is less restrictivethan those of Avallanedaet al. and Helsing.
IV. MULTIPHASE POLYCRYSTAL
An m-phase polycrystal could be first approximated byan n-component material of Sec. II,n5n11n21...1nm~each phasek51,...,m is composed ofnk components, eachcomponent consists of the crystals of the same orientation!and then letnk , n→`. Once the disorder-hypothesis~12!,~15!, and~18! is presumed, one can derive the bounds onsc
for a multiphase polycrystal from Eqs.~14! and~21! follow-ing the way of the previous section. Instead of Eq.~22!, inthis section the average value of a scalars-function S~s!would mean particularly
^S~s!&5 (a51
m
va^S~sa!&a ,
^S~sa!&a5 13@S~s1
a!1S~s2a!1S~s3
a!#,
wheres1a ,s2
a ,s3a are the principal conductivities of the phase
a ~a51,...,m!.The bounds forsc of a multiphase polycrystal are
P~2s0lm!<sc<P~2s0
um!, ~36!
whereP is defined in Eq.~24!;
s0um5min$s0u^s&<s0<smax, Qum~s0!<0%,
smax5max$s ia ; i51,2,3; a51,...,n%,
Qum~s0!5^~s2s0!@12^~s12s0!21&21~s12s0!
21#2&
19(a51
m
va^sa2s0&a^@12^~s12s0!21&21~sa
12s0!21#2&a ;
s0lm5max$s0usmin<s0<^s21&21, Qlm~s0!<0%.
smin5min$s ia ; i51,2,3; a51,...,n%,
Qlm~s0!5 K S 1s21
s0D F12 K 1s 1
1
2s0D 21L 21S 1s
11
2s0D 21G2)19(
a51
m
vaK 1
sa21
s0L
a
3 K F12 K S 1s 11
2s0D 21L 21
3S 1
sa 11
2s0D 21G2L
a
.
In case of an uniphase polycrystal, Eq.~36! is reduced toEqs.~28! and ~30!.
For multiphase polycrystals the principal conductivitiesof different phases may differ greatly, so the approximatebounds of Eq.~33!-type are not recommended.
In the case of isotropic phases, the bounds~36! reduce tothe ones for fully disordered~or perfectly random! multi-component materials,16,17 the latter lead to the known Mill-er’s bounds for two-component symmetric cell materials.14
Furthermore, there exists a simple relation between Miller’sG and our f 135 f 1/ f 3 @keeping in mind Eq. ~21!#:f 135(9G21)/(329G) ~see Ref. 17!. Like Miller’s bounds~with unspecifiedG between 1/3 and 1/9! and those ofHashin and Shtrikman,7 our bounds should be second orderin the series of expansion around homogeneity.
The bounds~34! are derived on the assumption ofdisorder-hypothesis~12!, ~15!, and ~18!. However, Eq.~18!may not be the natural one in the case of a multiphase poly-crystal: because of different conditions of formation, thecrystals from different phases may have quite different pat-terns of sizes and forms and Eq.~18! is no longer valid. Inthis case, one could not use Eq.~21!, but to come back toEqs. ~14! and ~3! @keeping Eqs.~12! and ~15!#. Looking atEq. ~3! we can see thatU~e!<0 when
s0>smax,
so from Eq.~14! we deduce
sc<P~2smax!, ~37!
P is defined in Eq.~24!. Similarly, the lower bound is
sc>P~2smin!. ~38!
The bounds~37! and ~38! are narrower than the arithmeticand harmonic mean bounds [P(0)<sc<P(`)] but less re-strictive than Eq.~36!. In case of an uniphase polycrystal,they reduce to the bounds given by Hashin and Shtrikman.
V. CONCLUSION
Based on the disorder-hypothesis~12!, ~15!, and ~18!,the new bounds for the effective conductivity of uniphasepolycrystals ~28!, ~29!, ~30!, and ~31! and of multiphasepolycrystals~36! have been established. The new bounds areexpected to apply to those polycrystals with random micro-structures. In the case of a multiphase polycrystal, the con-dition ~18! imposes a strong restriction on the microgeom-etry of the composite: the different phases should beinterchangeable in some sense. In the case of a random uni-phase polycrystal, the requirement seems natural becausetwo sets of crystals of different crystal orientations can inter-change their places and this is expected not to affect themacroscopic property. The mathematical method applied re-quires the phase boundary to be sufficiently smooth.
Our approach is based on the new variational principle@see Eq.~14!# which is a development of that of Hashin andShtrikman.7 The new variational inequalities contain an ad-ditional fluctuation term@seeU~e! in Eqs. ~14! and ~3!#,which makes it possible to reduce the uncertainty of theHashin–Shtrikman bounds.
More-complicated problems of elastic moduli will beconsidered in our upcombing article.20
2258 J. Appl. Phys., Vol. 80, No. 4, 15 August 1996 Pham Duc Chinh [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP:
130.70.241.163 On: Sun, 21 Dec 2014 15:24:18
1W. Voight, Lehrbuch der Krystallphysik~Teuber, Leipzig, 1928!.2A. Reuss, ZAMM9, 49 ~1929!.3R. Hill, Proc. Phys. Soc. London. Sect. A65, 349 ~1952!.4M. Avellaneda, A. V. Cherkaev, K. A. Lurie, and G. W. Milton, J. Appl.Phys.63, 4989~1988!.
5V. Nesi and G. W. Milton, J. Mech. Phys. Solids39, 525 ~1991!.6K. Schulgasser, J. Phys. C10, 407 ~1977!.7Z. Hashin and S. Shtrikman, Phys. Rev.130, 129 ~1963!.8J. Helsing, Proc. R. Soc. London Ser. A444, 363 ~1994!.9G. W. Milton and R. V. Kohn, J. Mech. Phys. Solids36, 597 ~1988!.10P. H. Dederichs and R. Zeller, Z. Phys.259, 103 ~1973!.
11L. J. Walpole, J. Mech. Phys. Solids14, 152 ~1966!.12R. M. Christensen,Mechanics of Composite Materials~Wiley, Chriches-ter, 1979!.
13D. C. Pham, Int. J. Eng. Sci.31, 11 ~1993!.14M. N. Miller, J. Math. Phys.10, 1988~1969!.15O. P. Bruno, Commun. Pure Appl. Math.XLIII , 769 ~1990!.16D. C. Pham, Arch. Rational Mech. Anal.127, 191 ~1994!.17D. C. Pham, Int. J. Solids Structures33, 1745~1996!.18J. Helsing, Proc. R. Soc. London Ser. A443, 451 ~1993!.19M. Avellaneda and J. Bruno, J. Math. Phys.31, 2047~1990!.20D. C. Pham~submitted, 1996!.
2259J. Appl. Phys., Vol. 80, No. 4, 15 August 1996 Pham Duc Chinh [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP:
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