Paper
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Mean Field Theory for Lossy Nonlinear Composites
Authors
K. W. Yu,Jerzy Tyszkiewicz
Osuolale Tiamiyu,Thomas Palmer,Gerald Lawson,Stephen Hill,Stanley Halvorsen,JULIAN ROMERO,Silvia Souza,Sylvain Fichet,Maribel Beltrán,Konstantinos Karafasoulis,Juan José Hernández-Rey,Andrew Round,Ivano Lippi,Carmela Fimognari,Anna Shcherbina,Galina Tereshchenko,Denis Kachanov,Chimit Ochirov,Walter Bravo,Irina Davydova,Mariya Zhestkova,José Gelonch Anyé,Natalia Migacheva,Svetlana Salugina,Victoria Voinova,Jorge Casaus,Osvaldo León Córdoba,Alexander Smirnov,Ilonka Guenther,Oliver Vitouch,Alexander Sokolov,Ekaterina Nikolaeva,Catherine Klein,Замира Раджабова,Tamari Maniya,Rodion Rasulov,Javier Berdugo,Римма Терлецкая,Rita Zoppi,Sang-Jin Sin,Augustin Kasser,Jonathan Ingram,Paolo Reggiani,Jochen Uebe,Abdelkader Dahchour,Alexey Merinov,D.Yu. Ovsyannikov,Patricia Mokhtarian,Nick Toms,Christis Hassapis,Giovanni Battista Luciani,Ana Poveda,Elham Afify,Andres Gomez-Caminero,George Sopko,Constantino Ledesma-Montes,SIMON KAYEMBA-KAY'S,HEATHER STANTON,Ilgiz Gareev,Erik Ole Jorgensen,Vladimír Porubčan,Rafal Ledzion,Marco Fabbrichesi,José Antonio Veira,Виктория Емельянова,stefan mehedinteanu,Charles Le Grand,Olga Senkevich,Carmen Teijeiro,Etsuji Okada,Shin Yuan Chen,Mikhail Akselrov,Juha Savola,George Thompson,Wesley Metzger,Imshik Lee,Aleksander Wittlin,Krishna Kumar,Joachim Mnich,Paul Martin,Hironobu Takahashi,George Leontaris,Emmanouella Remoundaki,Mokhtar Chmeissani,Jeffrey Templon,Fomina Boriskina,Eric Chamberod,Dmitry Lebedev,Elena Castro,Наиль Акрамов,Marina Shumikhina,Vyacheslav Lebedev,Natalia Zaikova,Darya Lakomova,Klaus Mehnert,Sanan Hamidov,Anastasia Ponasenko,Светлана Денисова,Светлана Шугаева,Reinaldo Ruggiero,mitsuhiro hirai,Zbigniew Rozwadowski,Javier Dominguez,Renat S. Аkchurin,Michael Hennecke,Indra K. 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arXiv:cond-mat/9711203v1 [cond-mat.soft] 20 Nov 1997
Mean Field Theory for Lossy Nonlinear Composites
K. W. Yu
Department of Physics, The Chinese University of Hong Kong,
Shatin, New Territories, Hong Kong
Abstract
The mean-field theory for lossy nonlinear composites, described by complex
and field-dependent dielectric functions, is presented. By using the spectral
representation of linear composites with identical microstructure, we develop
self-consistent equations for the effective response. We examine two types of
microstructure, namely, the Maxwell-Garnett approximation and the effective
medium approximation to illustrate the theory.
PACS Numbers: 72.20.Ht, 64.60.Ak, 72.60.+g
Typeset using REVTEX
1
Mean Field Theory for Lossy Nonlinear Composites
K. W. Yu
Department of Physics, The Chinese University of Hong Kong,
Shatin, New Territories, Hong Kong
Abstract
The mean-field theory for lossy nonlinear composites, described by complex
and field-dependent dielectric functions, is presented. By using the spectral
representation of linear composites with identical microstructure, we develop
self-consistent equations for the effective response. We examine two types of
microstructure, namely, the Maxwell-Garnett approximation and the effective
medium approximation to illustrate the theory.
PACS Numbers: 72.20.Ht, 64.60.Ak, 72.60.+g
Typeset using REVTEX
1
I. INTRODUCTION
Recently, we proposed a mean-field theory (MFT) for nonlinear composites [1]. The the-
ory was applied to nonlinear transport properties of conducting composites at zero frequency,
described by real conductivities and provided good agreement with numerical simulation [1].
However, for dielectric materials near relaxation, the loss may be important. It is tempting
to extend the MFT to complex and nonlinear permitivities. The local constitutive relation
of the composite system is given by
D = χ(E∗ · E)E, (1)
where χ = χ(x) is the (position dependent) complex third-order nonlinear coefficient. In
what follows, we denote |E|2 = E∗ · E and E2 = E · E for convenience. The effective response
is defined as the volume average of D
χeE2
0E0 = 1
V
∫
V
dV D(x), (2)
where V is the volume of composites and E0 is the applied (average) field and is taken to
be real:
E0 = 1
V
∫
V
dV E(x). (3)
The corresponding boundary-value problem of nonlinear composite media consists of the
field equations:
∇ · D = 0, (4)
∇ × E = 0. (5)
There exists a potential ϕ so that E = ∇ϕ. The boundary condition for ϕ is
ϕ = ϕ0 = E0 · x (6)
on surface (S) of the composite and ϕ0 is real. Consider the integral (1/V ) ∫
V dV D · E.
Through the field equations, we can convert it into an integral over the surface (S) of the
2
Recently, we proposed a mean-field theory (MFT) for nonlinear composites [1]. The the-
ory was applied to nonlinear transport properties of conducting composites at zero frequency,
described by real conductivities and provided good agreement with numerical simulation [1].
However, for dielectric materials near relaxation, the loss may be important. It is tempting
to extend the MFT to complex and nonlinear permitivities. The local constitutive relation
of the composite system is given by
D = χ(E∗ · E)E, (1)
where χ = χ(x) is the (position dependent) complex third-order nonlinear coefficient. In
what follows, we denote |E|2 = E∗ · E and E2 = E · E for convenience. The effective response
is defined as the volume average of D
χeE2
0E0 = 1
V
∫
V
dV D(x), (2)
where V is the volume of composites and E0 is the applied (average) field and is taken to
be real:
E0 = 1
V
∫
V
dV E(x). (3)
The corresponding boundary-value problem of nonlinear composite media consists of the
field equations:
∇ · D = 0, (4)
∇ × E = 0. (5)
There exists a potential ϕ so that E = ∇ϕ. The boundary condition for ϕ is
ϕ = ϕ0 = E0 · x (6)
on surface (S) of the composite and ϕ0 is real. Consider the integral (1/V ) ∫
V dV D · E.
Through the field equations, we can convert it into an integral over the surface (S) of the
2
composite: (1/V ) ∫
S dS · (Dϕ), which is equal to (1/V ) ∫
S dS · (Dϕ0) because ϕ = ϕ0 on
S. Converting the surface integral back to the volume integral, we obtain:
1
V
∫
V
dV D · E = 1
V
∫
V
dV D · E0. (7)
Hence we find an equivalent expression for the effective response:
χeE4
0 = 1
V
∫
V
dV D · E. (8)
Let us consider a lossy nonlinear composite in which inclusions of complex nonlinear coef-
ficient χ1, present at volume fraction p1, are randomly embedded in a host medium of χ2,
present at volume fraction p2, with the application of an external uniform field E0. Note
that p1 + p2 = 1. From Eqs.(1) and (8), we have therefore an expression for the effective
response:
χeE4
0 = p1χ1〈|E1|2E2
1〉 + p2χ2〈|E2|2E2
2〉, (9)
where
〈|Ei|2E2
i 〉 = 1
Vi
∫
Vi
dV |E|2E2, i = 1, 2. (10)
denotes the local field average within the ith component. Similar to the derivation of Eq.(7),
as ϕ∗ = ϕ0 on S, we arrive at
1
V
∫
V
dV D · E∗ = 1
V
∫
V
dV D · E0, (11)
which implies an alternative expression for the effective response:
χeE4
0 = p1χ1〈(|E1|2)2〉 + p2χ2〈(|E2|2)2〉, (12)
where
〈(|Ei|2)2〉 = 1
Vi
∫
Vi
dV (|E|2)2, i = 1, 2. (13)
We should remark that the different types of local field averages defined in Eqs.(10) and (13)
are generally not equal to each other. It is interesting to note that they both give the same
3
S dS · (Dϕ), which is equal to (1/V ) ∫
S dS · (Dϕ0) because ϕ = ϕ0 on
S. Converting the surface integral back to the volume integral, we obtain:
1
V
∫
V
dV D · E = 1
V
∫
V
dV D · E0. (7)
Hence we find an equivalent expression for the effective response:
χeE4
0 = 1
V
∫
V
dV D · E. (8)
Let us consider a lossy nonlinear composite in which inclusions of complex nonlinear coef-
ficient χ1, present at volume fraction p1, are randomly embedded in a host medium of χ2,
present at volume fraction p2, with the application of an external uniform field E0. Note
that p1 + p2 = 1. From Eqs.(1) and (8), we have therefore an expression for the effective
response:
χeE4
0 = p1χ1〈|E1|2E2
1〉 + p2χ2〈|E2|2E2
2〉, (9)
where
〈|Ei|2E2
i 〉 = 1
Vi
∫
Vi
dV |E|2E2, i = 1, 2. (10)
denotes the local field average within the ith component. Similar to the derivation of Eq.(7),
as ϕ∗ = ϕ0 on S, we arrive at
1
V
∫
V
dV D · E∗ = 1
V
∫
V
dV D · E0, (11)
which implies an alternative expression for the effective response:
χeE4
0 = p1χ1〈(|E1|2)2〉 + p2χ2〈(|E2|2)2〉, (12)
where
〈(|Ei|2)2〉 = 1
Vi
∫
Vi
dV (|E|2)2, i = 1, 2. (13)
We should remark that the different types of local field averages defined in Eqs.(10) and (13)
are generally not equal to each other. It is interesting to note that they both give the same
3
effective response [Eqs.(9) and (12)]. However, the nonlinear partial differential equations
pertaining to the boundary-value problem cannot be solved analytically. We shall invoke
the mean-field theory [1] to obtain an approximate expression for the effective response.
According to previous work [1], we consider the linear constitutive relation:
D = ǫE, (14)
where ǫ = ǫ(x) is the (position dependent) complex dielectric constant. By definition, we
find the effective linear response:
ǫeE0 = 1
V
∫
V
dV ǫE. (15)
From Eq.(7), we have therefore
ǫeE2
0 = p1ǫ1〈E2
1〉 + p2ǫ2〈E2
2〉. (16)
From Eq.(11), we have alternatively
ǫeE2
0 = p1ǫ1〈|E1|2〉 + p2ǫ2〈|E2|2〉. (17)
This expression will be useful for calculating 〈|Ei|2〉. In order to estimate χe, we invoke the
decoupling approximation [2]
〈|Ei|2E2
i 〉 = 〈|Ei|2〉〈E2
i 〉, i = 1, 2. (18)
within the ith component. This assumption is good in microstructure for which the local
electric field is nearly uniform within the ith component, but less accurate when these
fluctuations are large, as in a random composite near the percolation threshold [2]. With
this assumption,
χeE4
0 = p1χ1〈|E1|2〉〈E2
1〉 + p2χ2〈|E2|2〉〈E2
2〉. (19)
Comparing Eqs.(16) and (19), it is tempting to write ǫ1 = χ1〈|E1|2〉, ǫ2 = χ2〈|E2|2〉 and
ǫe = χeE2
0. Hence, within the decoupling approximation, we may interpret the nonlinear
4
pertaining to the boundary-value problem cannot be solved analytically. We shall invoke
the mean-field theory [1] to obtain an approximate expression for the effective response.
According to previous work [1], we consider the linear constitutive relation:
D = ǫE, (14)
where ǫ = ǫ(x) is the (position dependent) complex dielectric constant. By definition, we
find the effective linear response:
ǫeE0 = 1
V
∫
V
dV ǫE. (15)
From Eq.(7), we have therefore
ǫeE2
0 = p1ǫ1〈E2
1〉 + p2ǫ2〈E2
2〉. (16)
From Eq.(11), we have alternatively
ǫeE2
0 = p1ǫ1〈|E1|2〉 + p2ǫ2〈|E2|2〉. (17)
This expression will be useful for calculating 〈|Ei|2〉. In order to estimate χe, we invoke the
decoupling approximation [2]
〈|Ei|2E2
i 〉 = 〈|Ei|2〉〈E2
i 〉, i = 1, 2. (18)
within the ith component. This assumption is good in microstructure for which the local
electric field is nearly uniform within the ith component, but less accurate when these
fluctuations are large, as in a random composite near the percolation threshold [2]. With
this assumption,
χeE4
0 = p1χ1〈|E1|2〉〈E2
1〉 + p2χ2〈|E2|2〉〈E2
2〉. (19)
Comparing Eqs.(16) and (19), it is tempting to write ǫ1 = χ1〈|E1|2〉, ǫ2 = χ2〈|E2|2〉 and
ǫe = χeE2
0. Hence, within the decoupling approximation, we may interpret the nonlinear
4
component as a linear dielectric material with a field-dependent dielectric constant. Suppose
ǫe is known as a function of its constituent dielectric constants,
ǫe = F (ǫ1, ǫ2, p1), (20)
then we find the local field averages [1]
p1〈E2
1〉 = ∂ǫe
∂ǫ1
E2
0, (21)
p2〈E2
2〉 = ∂ǫe
∂ǫ2
E2
0. (22)
However, we have to determine 〈|Ei|2〉 in the ith component, which is generally not equal
to 〈E2
i 〉. Thus a straightforward application of the previous formalism [1] is impossible. We
resort to an alternative approach based on the spectral representation [3].
In what follows, we shall consider two important types of microstructures: (1) Dispersion
microstructures as in the Maxwell-Garnett approximation [4] and (2) symmetric microstruc-
tures as in the Bruggeman effective medium approximation [5].
II. MAXWELL-GARNETT APPROXIMATION
The Maxwell-Garnett approximation (MGA) is good for dispersion microstructures and
the theory is inherently non-symmetrical [4]. For convenience, we consider the case in which
component 1 is embedded in component 2. For a linear composite of ǫ1 and ǫ2 at volume
fractions p1 and p2 respectively, the MGA reads [4]:
ǫe − ǫ2
ǫe + (d − 1)ǫ2
= p1
ǫ1 − ǫ2
ǫ1 + (d − 1)ǫ2
, (23)
where d is the dimensionality of composites. Solving Eq.(23), we obtain
ǫe = ǫ2[ǫ1(dp1 + p2) + ǫ2(d − 1)p2]
p2ǫ1 + (d − p2)ǫ2
. (24)
From Eqs.(21) and (22), we can calculate the local field averages:
p1〈E2
1〉 = p1d2ǫ2
2
[p2ǫ1 + (d − p2)ǫ2]2 E2
0. (25)
5
ǫe is known as a function of its constituent dielectric constants,
ǫe = F (ǫ1, ǫ2, p1), (20)
then we find the local field averages [1]
p1〈E2
1〉 = ∂ǫe
∂ǫ1
E2
0, (21)
p2〈E2
2〉 = ∂ǫe
∂ǫ2
E2
0. (22)
However, we have to determine 〈|Ei|2〉 in the ith component, which is generally not equal
to 〈E2
i 〉. Thus a straightforward application of the previous formalism [1] is impossible. We
resort to an alternative approach based on the spectral representation [3].
In what follows, we shall consider two important types of microstructures: (1) Dispersion
microstructures as in the Maxwell-Garnett approximation [4] and (2) symmetric microstruc-
tures as in the Bruggeman effective medium approximation [5].
II. MAXWELL-GARNETT APPROXIMATION
The Maxwell-Garnett approximation (MGA) is good for dispersion microstructures and
the theory is inherently non-symmetrical [4]. For convenience, we consider the case in which
component 1 is embedded in component 2. For a linear composite of ǫ1 and ǫ2 at volume
fractions p1 and p2 respectively, the MGA reads [4]:
ǫe − ǫ2
ǫe + (d − 1)ǫ2
= p1
ǫ1 − ǫ2
ǫ1 + (d − 1)ǫ2
, (23)
where d is the dimensionality of composites. Solving Eq.(23), we obtain
ǫe = ǫ2[ǫ1(dp1 + p2) + ǫ2(d − 1)p2]
p2ǫ1 + (d − p2)ǫ2
. (24)
From Eqs.(21) and (22), we can calculate the local field averages:
p1〈E2
1〉 = p1d2ǫ2
2
[p2ǫ1 + (d − p2)ǫ2]2 E2
0. (25)
5
p2〈E2
2〉 =
(
1 − p1d[dǫ2
2 − p2(ǫ1 − ǫ2)2]
[p2ǫ1 + (d − p2)ǫ2]2
)
E2
0. (26)
In MGA, the local field in the inclusion is uniform, we can determine E1 and hence E∗
1
explicitly, we find
p1〈|E1|2〉 = p1d2|ǫ2|2
|p2ǫ1 + (d − p2)ǫ2|2 E2
0. (27)
From Eq.(17) and using Eqs.(24) and (27), we find
p2〈|E2|2〉 =
(
1 − p1d[d|ǫ2|2 − p2|ǫ1 − ǫ2|2]
|p2ǫ1 + (d − p2)ǫ2|2
)
E2
0. (28)
We should remark that although |〈E2
1〉| = 〈|E1|2〉 in MGA, |〈E2
2〉| is generally not equal
to 〈|E2|2〉. In fact, |〈E2
2〉| ≤ 〈|E2|2〉. A common error in the literature is to treat these
averages equal even when χ is complex. If we write ǫ1 = χ1〈|E1|2〉 and ǫ2 = χ2〈|E2|2〉, then
Eqs.(27)–(28) can be solved self-consistently for 〈|E1|2〉 and 〈|E2|2〉 and hence the effective
nonlinear response can be calculated.
In Fig.1, we present the MGA results in three dimensions (3D). We let χ1 = 1 + 3i
and χ2 = 3 + i and compute χe as a function of p1. As p1 increases, Re(χe) decreases from
Re(χ2) = 3 towards Re(χ1) = 1 while Im(χe) increases from Im(χ2) = 1 towards Im(χ1) = 3.
Since the contrast between the two components is relatively small, the local field averages
〈|E1|2〉 and 〈|E2|2〉 remain close to unity. However, unlike |〈E2
1〉|, |〈E2
2〉| shows a significant
deviation from 〈|E2|2〉, indicating the nonuniformity of local field in the host.
For two-component composites, it has proved convenient to adopt the spectral represen-
tation of the effective linear response [3]: Let v = 1 − ǫ1/ǫ2, w = 1 − ǫe/ǫ2, and s = 1/v, we
find
w(s) =
∫ 1
0
m(s′)ds′
s − s′ , (29)
where m(s) is the spectral density which is obtained through a limiting process:
m(s) = lim
η→0+ − 1
π Im w(s + iη). (30)
Furthermore, m(s) obeys the sum rule
6
2〉 =
(
1 − p1d[dǫ2
2 − p2(ǫ1 − ǫ2)2]
[p2ǫ1 + (d − p2)ǫ2]2
)
E2
0. (26)
In MGA, the local field in the inclusion is uniform, we can determine E1 and hence E∗
1
explicitly, we find
p1〈|E1|2〉 = p1d2|ǫ2|2
|p2ǫ1 + (d − p2)ǫ2|2 E2
0. (27)
From Eq.(17) and using Eqs.(24) and (27), we find
p2〈|E2|2〉 =
(
1 − p1d[d|ǫ2|2 − p2|ǫ1 − ǫ2|2]
|p2ǫ1 + (d − p2)ǫ2|2
)
E2
0. (28)
We should remark that although |〈E2
1〉| = 〈|E1|2〉 in MGA, |〈E2
2〉| is generally not equal
to 〈|E2|2〉. In fact, |〈E2
2〉| ≤ 〈|E2|2〉. A common error in the literature is to treat these
averages equal even when χ is complex. If we write ǫ1 = χ1〈|E1|2〉 and ǫ2 = χ2〈|E2|2〉, then
Eqs.(27)–(28) can be solved self-consistently for 〈|E1|2〉 and 〈|E2|2〉 and hence the effective
nonlinear response can be calculated.
In Fig.1, we present the MGA results in three dimensions (3D). We let χ1 = 1 + 3i
and χ2 = 3 + i and compute χe as a function of p1. As p1 increases, Re(χe) decreases from
Re(χ2) = 3 towards Re(χ1) = 1 while Im(χe) increases from Im(χ2) = 1 towards Im(χ1) = 3.
Since the contrast between the two components is relatively small, the local field averages
〈|E1|2〉 and 〈|E2|2〉 remain close to unity. However, unlike |〈E2
1〉|, |〈E2
2〉| shows a significant
deviation from 〈|E2|2〉, indicating the nonuniformity of local field in the host.
For two-component composites, it has proved convenient to adopt the spectral represen-
tation of the effective linear response [3]: Let v = 1 − ǫ1/ǫ2, w = 1 − ǫe/ǫ2, and s = 1/v, we
find
w(s) =
∫ 1
0
m(s′)ds′
s − s′ , (29)
where m(s) is the spectral density which is obtained through a limiting process:
m(s) = lim
η→0+ − 1
π Im w(s + iη). (30)
Furthermore, m(s) obeys the sum rule
6
∫ 1
0
m(s′)ds′ = p1. (31)
Eqs.(24)–(28) can readily be converted into the spectral representation. We find
w = p1
s − s1
, (32)
where s1 = p2/d. From Eq.(30), we obtain the spectral density in MGA:
m(s) = p1δ(s − s1). (33)
Similarly, the local field averages [Eqs.(25)–(28)] are given by
〈E2
1〉 = E2
0
(1 − vs1)2 , (34)
〈|E1|2〉 = E2
0
|1 − vs1|2 , (35)
p2〈E2
2〉 =
(
1 − p1(1 − v2s1)
(1 − vs1)2
)
E2
0, (36)
p2〈|E2|2〉 =
(
1 − p1(1 − |v|2s1)
|1 − vs1|2
)
E2
0. (37)
If we let v = 1 − χ1〈|E1|2〉/χ2〈|E2|2〉 and solve Eqs.(35) and (37) self-consistently for 〈|E1|2〉
and 〈|E2|2〉, the effective nonlinear response can be determined. Although the solution
appears somewhat simpler, the same results are obtained.
III. EFFECTIVE MEDIUM APPROXIMATION
The Bruggeman effective-medium approximation (EMA) is known to be symmetrical
with respect to interchanging components 1 and 2 [5]. Note that there is a percolation
threshold in the theory. For a random linear composite of ǫ1 and ǫ2 at volume fractions p1
and p2 respectively, the self-consistency equations reads [5]:
p1
ǫ1 − ǫe
ǫ1 + (d − 1)ǫe
+ p2
ǫ2 − ǫe
ǫ2 + (d − 1)ǫe
= 0, (38)
where d is the dimensionality of composites. Solving Eq.(38), we obtain ǫe and hence w and
the spectral density
7
0
m(s′)ds′ = p1. (31)
Eqs.(24)–(28) can readily be converted into the spectral representation. We find
w = p1
s − s1
, (32)
where s1 = p2/d. From Eq.(30), we obtain the spectral density in MGA:
m(s) = p1δ(s − s1). (33)
Similarly, the local field averages [Eqs.(25)–(28)] are given by
〈E2
1〉 = E2
0
(1 − vs1)2 , (34)
〈|E1|2〉 = E2
0
|1 − vs1|2 , (35)
p2〈E2
2〉 =
(
1 − p1(1 − v2s1)
(1 − vs1)2
)
E2
0, (36)
p2〈|E2|2〉 =
(
1 − p1(1 − |v|2s1)
|1 − vs1|2
)
E2
0. (37)
If we let v = 1 − χ1〈|E1|2〉/χ2〈|E2|2〉 and solve Eqs.(35) and (37) self-consistently for 〈|E1|2〉
and 〈|E2|2〉, the effective nonlinear response can be determined. Although the solution
appears somewhat simpler, the same results are obtained.
III. EFFECTIVE MEDIUM APPROXIMATION
The Bruggeman effective-medium approximation (EMA) is known to be symmetrical
with respect to interchanging components 1 and 2 [5]. Note that there is a percolation
threshold in the theory. For a random linear composite of ǫ1 and ǫ2 at volume fractions p1
and p2 respectively, the self-consistency equations reads [5]:
p1
ǫ1 − ǫe
ǫ1 + (d − 1)ǫe
+ p2
ǫ2 − ǫe
ǫ2 + (d − 1)ǫe
= 0, (38)
where d is the dimensionality of composites. Solving Eq.(38), we obtain ǫe and hence w and
the spectral density
7
m(s) = dp1 − 1
d − 1 δ(s)θ(dp1 − 1) + m1(s), (39)
where δ(s) and θ(dp1 − 1) denote the Dirac delta function and the Heavyside step function
respectively, and
m1(s) = d
2π(d − 1)s
√
(s − s1)(s2 − s), s1 < s < s2, (40)
= 0, otherwise,
where s1 and s2 are given by
s1 = 1
d
(
1 + (d − 2)p1 − 2
√
(d − 1)p1(1 − p1)
)
,
s2 = 1
d
(
1 + (d − 2)p1 + 2
√
(d − 1)p1(1 − p1)
)
. (41)
When p1 > 1/d, which is the percolation threshold in EMA, component 1 percolates the
composite and there is a δ function contribution to m(s) at s = 0. Eqs.(34)–(37) can readily
be generalized to [6]:
p1〈E2
1〉 =
∫ 1
0
m(s′)ds′
(1 − vs′)2 E2
0, (42)
p1〈|E1|2〉 =
∫ 1
0
m(s′)ds′
|1 − vs′|2 E2
0, (43)
p2〈E2
2〉 =
(
1 −
∫ 1
0
(1 − v2s′)m(s′)ds′
(1 − vs′)2
)
E2
0, (44)
p2〈|E2|2〉 =
(
1 −
∫ 1
0
(1 − |v|2s′)m(s′)ds′
|1 − vs′|2
)
E2
0. (45)
One can check that when the MGA spectral density [Eq.(33)] is used, Eqs.(34)–(37) are
recovered. Again we let v = 1 − χ1〈|E1|2〉/χ2〈|E2|2〉. Eqs.(43) and (45) are coupled integral
equations. They can be solved numerically for 〈|E1|2〉 and 〈|E2|2〉 and hence the effective
nonlinear response can be determined.
In Fig.2, we present the EMA results in 3D. We let χ1 = 1 + 3i and χ2 = 3 + i. Fig.2
exhibits similar behavior as Fig.1, however, with some differences. Here Re(χe) and Im(χe)
cross at p1 = 0.5 because EMA describes the symmetric microstructure and we have the
8
d − 1 δ(s)θ(dp1 − 1) + m1(s), (39)
where δ(s) and θ(dp1 − 1) denote the Dirac delta function and the Heavyside step function
respectively, and
m1(s) = d
2π(d − 1)s
√
(s − s1)(s2 − s), s1 < s < s2, (40)
= 0, otherwise,
where s1 and s2 are given by
s1 = 1
d
(
1 + (d − 2)p1 − 2
√
(d − 1)p1(1 − p1)
)
,
s2 = 1
d
(
1 + (d − 2)p1 + 2
√
(d − 1)p1(1 − p1)
)
. (41)
When p1 > 1/d, which is the percolation threshold in EMA, component 1 percolates the
composite and there is a δ function contribution to m(s) at s = 0. Eqs.(34)–(37) can readily
be generalized to [6]:
p1〈E2
1〉 =
∫ 1
0
m(s′)ds′
(1 − vs′)2 E2
0, (42)
p1〈|E1|2〉 =
∫ 1
0
m(s′)ds′
|1 − vs′|2 E2
0, (43)
p2〈E2
2〉 =
(
1 −
∫ 1
0
(1 − v2s′)m(s′)ds′
(1 − vs′)2
)
E2
0, (44)
p2〈|E2|2〉 =
(
1 −
∫ 1
0
(1 − |v|2s′)m(s′)ds′
|1 − vs′|2
)
E2
0. (45)
One can check that when the MGA spectral density [Eq.(33)] is used, Eqs.(34)–(37) are
recovered. Again we let v = 1 − χ1〈|E1|2〉/χ2〈|E2|2〉. Eqs.(43) and (45) are coupled integral
equations. They can be solved numerically for 〈|E1|2〉 and 〈|E2|2〉 and hence the effective
nonlinear response can be determined.
In Fig.2, we present the EMA results in 3D. We let χ1 = 1 + 3i and χ2 = 3 + i. Fig.2
exhibits similar behavior as Fig.1, however, with some differences. Here Re(χe) and Im(χe)
cross at p1 = 0.5 because EMA describes the symmetric microstructure and we have the
8
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