Effective conductivities of two-phase composites with a singular phase
JOURNAL OF APPLIED PHYSICS 105, 103503 共2009兲
Effective conductivities of two-phase composites with a singular phase Liping Liua兲 Department of Mechanical Engineering, University of Houston, Houston, Texas 77204, USA
共Received 17 January 2009; accepted 3 March 2009; published online 18 May 2009兲 We calculate the effective conductivity of a two-phase composite with a periodic array of inhomogeneities. The shape of the inhomogeneities is assumed to be a periodic E-inclusion. The effective conductivity is expressed in terms of the volume fraction of the inhomogeneities and a matrix, which characterizes the shape of the periodic E-inclusion. This solution is rigorous, closed-form, and applicable to situations that the conductivity of the inhomogeneities is singular, i.e., zero or infinite. Further, when the periodic E-inclusion degenerates to a periodic array of slits with vanishing volume fraction, we give explicit solutions to local fields and effective conductivity of the composite with singular inhomogeneities. © 2009 American Institute of Physics. 关DOI: 10.1063/1.3110026兴 I. INTRODUCTION
High-contrast composites have wide industrial applications. Examples include metal and glass foams, fibrous metal and glass materials, fiber-reinforced materials, and fractured porous media. Once the notion of effective properties of composites or heterogeneous media is established, we are faced with the critical problem of finding the effective properties of a given composite and, if possible, relating the effective properties with the microstructure of the composite. However this is difficult even for composites with simple microstructures, for example, a two-phase composite with a periodic array of inhomogeneous spheres. In this and similar cases, heuristic methods such as the effective medium theory 共Ref. 1 and references therein兲 and the self-consistent method2 can give us closed-form formulas of the effective properties. These approximate formulas are in good agreement with experiments when the contrast between the two phases is small but unsatisfactory when the contrast of the constituent phases becomes large. If we idealize the physical properties of the inhomogeneities to be infinite or zero, many of the approximate formulas even yield unphysical results. Further, rigorous bounds such as the Hashin–Shtrikman bounds3 are not of much use since they are far apart from each other for high-contrast composites. So it is important for both theory and application to have a reasonable estimate of the effective properties of high-contrast composites. This is the purpose of this paper. Theoretical results are available for a small number of particular cases in the literature. Keller4 considered a composite of a cubic array of perfectly conducting spheres embedded in a normal conducting medium and obtained an asymptotic formula for the effective conductivity when the gaps between nearby spheres are small. Dykne5 showed that a two-phase system of insulating and conducting materials could undergo a dielectrics-conductor transition at certain critical concentration. For random high-contrast composites, a number of authors6,7 used a discrete network to model a high-contrast composite based on physical consideration. a兲
Electronic mail: [email protected].
0021-8979/2009/105共10兲/103503/9/$25.00
Kozlov8 and Berlyand and Kolpakov9 showed that the discrete network is a sound model of the original continuum problem. From these works, we observe that high-contrast composites have two distinguishing features compared with normal composites. The first is that high-contrast composites can undergo a percolation transition, and the second is that the effective properties are dependent on the local geometry of the composite, say, the interinhomogeneity distance, as much as the global parameter, such as the volume fraction. In particular, if the physical property of the inhomogeneities is zero or infinite, the inhomogeneities could have a significant impact on the effective properties even if their volume fraction vanishes. This is well understood in the context of fracture mechanics10,11 but seems unnoticed for conductive composites. In this paper we give explicit solutions to the effective properties of a class of periodic composites. To obtain these solutions we choose a special class of microstructures called periodic E-inclusions or Vigdergauz12 structures in two dimensions.13 Unlike previous examples, our solutions are rigorous and closed-form, which express the effective properties in terms of the volume fraction of the inhomogeneities and the matrix Q, which characterizes the shape of the inhomogeneities. A disadvantage of our solutions is that the shape of periodic E-inclusions is not directly given; we need to solve a variational inequality to find a periodic E-inclusion.13,14 Nevertheless, without solving the variational inequality we know qualitatively the shapes of periodic E-inclusions and how they depend on the volume fraction and the shape matrix Q. In particular, when the shape matrix Q is singular and the volume fraction approaches to zero, the periodic E-inclusion degenerates to a slit in two dimensions. In this case we express the effective properties in terms of the length of the slit. It indicates that the widely used rules of mixtures, which interpolate between the properties of the matrix and the inhomogeneities by volume fractions, could be qualitatively misleading for high-contrast composites. Two remarks are in order here about the scope of this paper. For simplicity we discuss the conductivity problem,
105, 103503-1
© 2009 American Institute of Physics
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103503-2
J. Appl. Phys. 105, 103503 共2009兲
Liping Liu
but note that the solutions are applicable to other physical properties including dielectric properties, diffusive properties, transport properties, and at least qualitatively, elastic properties. More, our solutions are for periodic composites. Therefore, the effects of percolation and randomness are not addressed here. The paper is organized as follows. In Sec. II we formulate the governing equation for a periodic composite and derive the formulas for calculating the effective properties of the composite. In Sec. III we define the periodic E-inclusion and show how to calculate the effective properties of a composite with a periodic E-inclusion microstructure. In Sec. IV we discuss a particular situation that the periodic E-inclusion degenerates to a periodic array of slits. For this particular situation we give explicit solutions to local fields and effective properties for a rectangular lattice in Sec. IV A and for a rhombic lattice in Sec. IV B. Finally we summarize our results in Sec. V. II. PROBLEM FORMULATION
Let Y 傺 IRn be a unit cell associated with a Bravais lattice L, ⍀ 傺 Y be an inclusion containing an inhomogeneity, and be the volume fraction of the inhomogeneities. Consider a periodic two-phase composite with conductivity given by A共x兲 =
再
k0I if x 苸 Y \ ⍀ k1I if x 苸 ⍀,
冎
共1兲
where k0 ⬎ 0, k1 ⱖ 0, and I 苸 IRn⫻n is the identity matrix. For ease of terminology, we refer to phase-0 as the matrix and phase-1 as the inhomogeneities. From the homogenization theory,15 the effective conductivity of the composite, described by a symmetric tensor Ae, is given by f · Aef = min u苸W
W
共ⵜu + f兲 · A共x兲共ⵜu + f兲,
共2兲
Y
where WV = 共1 / 兩V兩兲兰V denotes the average of the integrand over V 共兩V兩 denotes the volume of V兲, f 苸 IRn is the average applied field, and the admissible space W is the collection of all periodic square integrable functions u : IRn → IR whose gradients remain square integrable. To evaluate the effective conductivity tensor Ae, we need to solve the Euler–Lagrange equation of Eq. 共2兲 for the minimizer uf,
再
div关A共x兲共ⵜuf + f兲兴 = 0 on Y , periodic boundary conditions on Y ,
冎
共3兲
and then compute the integral for u = uf on the right hand side of Eq. 共2兲. The effective conductivity of a composite can be expressed as a boundary integral of the minimizer uf or an integral on one of the phases. To see this, we notice uf being a periodic function and A共x兲共ⵜuf + f兲 being divergence-free imply
冕
ⵜuf = 0
and
Y
Therefore, we have
冕
Y
ⵜuf · A共x兲共ⵜuf + f兲 = 0.
共4兲
f · A ef =
W W
共ⵜuf + f兲 · A共x兲共ⵜuf + f兲
Y
=
f · A共x兲共ⵜuf + f兲 = f ·
Y
=
1 兩Y兩
冕
Y
W
共ⵜx兲A共x兲共ⵜuf + f兲
Y
共f · x兲n · A共x兲共ⵜuf + f兲,
共5兲
where n is the outward normal on Y. Alternatively, from the first line in Eq. 共5兲 we find f · A ef = k 0
W
W W
f · 共ⵜuf + f兲 + 共k1 − k0兲f ·
Y
冋
= k0兩f兩2 + 共k1 − k0兲f · f − 共1 − 兲
⍀
共ⵜuf + f兲
Y\⍀
册
共ⵜuf + f兲 . 共6兲
For general inclusions, we do not have a closed-form solution of Eq. 共2兲, and thus much of the works have been focused on the bounds on the effective conductivity tensor Ae and numerical methods that compute Ae for a given inclusion ⍀. An exception is the case that the inclusion ⍀ is a periodic E-inclusion. Below we describe what a periodic E-inclusion is and calculate the effective conductivity of a composite with the inclusion being a periodic E-inclusion.
III. EFFECTIVE CONDUCTIVITIES OF COMPOSITES WITH PERIODIC E-INCLUSION INHOMOGENEITIES
Recently, the Liu et al.16 found a class of special inclusions for which a closed-form solution of Eq. 共2兲 is available. These special inclusions, termed as periodic E-inclusions, are defined as an inclusion ⍀ 傺 Y such that the overdetermined problem
冦
⌬ = − ⍀ on Y , ⵜⵜ = − 共1 − 兲Q on ⍀, periodic boundary conditions on Y
冧
共7兲
admits a solution where ⍀ is the characteristic function of ⍀ and Q 苸 Q ª 兵M n⫻n :M is positive semidefinite with Tr共M兲 苸 IRsym
= 1其.
共8兲
The Liu et al.13 showed the existence of periodic E-inclusions for any volume fraction 苸 共0 , 1兲, any matrix Q 苸 Q, and any Bravais lattice L. Figure 1 shows three examples of periodic E-inclusions in two dimensions 共see Refs. 17, 16, and 13 for more examples兲. We remark that periodic E-inclusions are generalizations of two-dimensional structures constructed by Vigdergauz.12 If the inclusion ⍀ for the composite is a periodic E-inclusion with matrix Q and volume fraction , we claim that a solution of Eq. 共3兲 is given by
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1
Ae = k0I + 共k1 − k0兲I − 共1 − 兲共k1 − k0兲2Q关共1 − 兲共k1 − k0兲Q + k0I兴−1 .
0.8
共13兲
We remark that the above formula is rigorous and attains the lower 共upper兲 Hashin–Shtrikman3 bound if k0 ⬍ k1 共k0 ⬎ k1兲. It has a few regimes that need separate attention.
0.6 0.4
共1兲 If k1 is nonsingular, i.e., ⫽0 or +⬁, formula 共13兲 gives a definite answer to the effective conductivity tensor of the composite of a periodic E-inclusion with any matrix Q 苸 Q and any volume fraction 苸 关0 , 1兴 since 共1 − 兲 ⫻共k1 − k0兲Q + k0I is invertible. In particular, if we assume the composite is isotropic, then the corresponding matrix Q is equal to I / n. Denoting by ke the isotropic conductivity of the composite, from Eq. 共13兲 we have
0.2 0 −0.2 −0.4 −0.6
ke 共1 − 兲共n − 1兲 + 关1 + 共n − 1兲兴k1/k0 = . k0 n − 1 + + 共1 − 兲k1/k0
−0.8 −1 −1
−0.5
0
0.5
1
FIG. 1. 共Color online兲 From outward to inward, the regions bounded by the curves are periodic E-inclusions with matrices Q and volume fractions given by Eq. 共16兲 and unit cell Y = 共−1 , 1兲2.
共2兲 If Q is positive definite, formula 共13兲 also gives a definite answer even if k1 is singular. In particular, we have Ae/k0 = 共1 − 兲I − 共1 − 兲Q关− 共1 − 兲Q + I兴−1 if k1 = 0, Ae/k0 = I +
uf = a · ⵜ,
a = 共k1 − k0兲关共1 − 兲共k1 − k0兲Q + k0I兴−1f. 共9兲
To see this, we notice that for Eq. 共3兲 the interfacial condition on ⍀ can be written as k1n · 关ⵜuf共x−兲 + f兴 = k0n · 关ⵜuf共x+兲 + f兴
∀ x 苸 ⍀, 共10兲
whereas being a solution of ⌬ = − ⍀ satisfies ⵜⵜ共x−兲 − ⵜⵜ共x+兲 = − n 丢 n
∀ x 苸 ⍀,
冋
共1 − 兲Q + =
= k1n · 关− 共1 − 兲Qa + f兴 − k1关− 共1 − 兲n · Qa + n · f兴 + k0关− 共1 − 兲n · Qa + a · n + n · f兴 = k1n · 关ⵜa · ⵜ共x−兲 + f兴 + n · 关共k1 − k0兲共1 − 兲Qa + k0a + 共k0 − k1兲f兴. 共12兲 Therefore, if we choose the vector a as in Eq. 共9兲, the last term on the right hand side of Eq. 共12兲 vanishes, and hence uf = a · ⵜ satisfies the interfacial condition 共10兲. Further, we can easily verify that uf = a · ⵜ satisfies the first of Eq. 共3兲 on the interior and exterior of ⍀ and the periodic boundary conditions on Y. We thus conclude that uf = a · ⵜ is a solution of Eq. 共3兲 if ⍀ is a periodic E-inclusion and is given by Eq. 共7兲. Using Eq. 共9兲 we can calculate the effective conductivity tensor Ae for a periodic E-inclusion. By the first line of Eq. 共6兲 and the second of Eq. 共7兲 we find
共15兲
k0 I k1 − k0
册
−1
1 k0 Q−1Q−1 Q−1 − 2 1− 共1 − 兲 共k1 − k0兲
共11兲
k0n · 关ⵜa · ⵜ共x+兲 + f兴
Q−1 if k1 = + ⬁. 1−
The second of the above formula follows from the fact that as k1 → + ⬁,
+O
where n is the outward normal on ⍀ and x− 共x + 兲 denotes the limit from the inside 共outside兲 of ⍀. From Eq. 共11兲 and the second of Eq. 共7兲, direct calculation reveals that for any a 苸 IRn,
共14兲
冉冏
k0 k1 − k0
冏冊 2
.
If the composite is assumed to be isotropic with the effective conductivity denoted by ke0 共k⬁e 兲 for k1 = 0 共k1 = + ⬁兲, then from Eq. 共14兲 or Eq. 共15兲 we have ke0 共1 − 兲共n − 1兲 = , k0 n−1+
k⬁e 1 + 共n − 1兲 = . k0 1−
共3兲 If 苸 共0 , 1兴, Q is singular, and k1 = 0 or +⬁, formula 共15兲 yields a definite answer to Ae if we interpret the inverse of a matrix M 苸 IRn⫻n as M−1 = lim 关M + I兴−1 . 0
共4兲 If = 0 and Q is singular, formula 共15兲 is not necessarily meaningful since we could get a term such as 0 · ⬁. Nevertheless, such situations are physically interesting. Below we focus on this regime in two dimensions.
IV. EFFECTIVE CONDUCTIVITIES OF COMPOSITES WITH PERIODIC SLIT INHOMOGENEITIES
In this section we restrict ourselves to two dimensions 共n = 2兲. From the viewpoint of last section and in connection with geometry, we are interested in the following limit: we
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103503-4
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fix the two ends of the periodic E-inclusion ⍀ along x1-direction and let the dimension of ⍀ along x2-direction shrink to zero. Using formulas in Refs. 17 and 16, we plot three periodic E-inclusions in two dimensions for unit cell Y = 共−1 , 1兲2 in Fig. 1. From outward to inward, the matrix Q and volume fraction are given by
Q=
冤
r 1+r
0
0
1 1+r
冥
,
r = 0.5, 0.2, 0.01
and
FIG. 2. 共Color online兲 Rectangular unit cell.
共16兲
= 0.42, 0.17, 0.04. We note that as Q→
冋 册 0 0 0 1
and
㜷共z兩1, 3兲 =
→ 0,
the periodic E-inclusion approaches to a slit parallel to x1-axis. Our goal is to compute the effective conductivity Ae in this limit for singular k1, for which Eq. 共13兲 or Eq. 共15兲 is no longer applicable. If k1 is a finite positive number, from Eq. 共13兲 we have Ae = k0I, as expected. To proceed, we reformulate problem 共3兲 for k1 = + ⬁ or 0. If k1 = + ⬁, the inclusion ⍀ necessarily remains as an equipotential body, and so problem 共3兲 is equivalent to
冦
¯, ⌬uf = 0 on Y \ ⍀ t · 共ⵜuf + f兲 = 0 on ⍀, periodic boundary conditions on Y ,
冧
共17兲
where t is the tangent on ⍀. Since ⍀ is a slit parallel to x1-axis, the second of Eq. 共17兲 can be rewritten as
uf共x1,x2兲 + f1 = 0 x1
∀ 共x1,x2兲 苸 ⍀.
共18兲
If k1 = 0, from the Gaussian theorem we infer that problem 共3兲 is equivalent to
冦
¯, ⌬uf = 0 on Y \ ⍀ n · 共ⵜuf + f兲 = 0 on ⍀, periodic boundary conditions on Y .
冧
共19兲
∀ 共x1,x2兲 苸 ⍀.
L = 兵211 + 223:1, 2 苸 Z其
共21兲
be the lattice. Associated with this lattice, we denote by Y the open parallelogram with vertices 0, 21, 22 = 2共1 + 3兲, and 23 and recall that the Weierstrass 㜷-function
共22兲
A. Rectangular unit cell
We first assume that 1 = ␣, 3 = i 共␣ ,  ⬎ 0兲 and that the slit ⍀ = 兵x1 + i : ␣ − ␣0 ⬍ x1 ⬍ ␣ + ␣0其 共0 ⬍ ␣0 ⬍ ␣兲 lies on the horizontal line Im关z兴 = . In this case, Y is an open rectangular with base 2␣ and height 2 共see Fig. 2兲. We briefly write 㜷共z兲 = 㜷共z 兩 1 = ␣ , 3 = i兲 in this section. From Markushevich18 we know that 㜷共z兲 takes real values on the vertical lines Re关z兴 = ␣ and horizontal lines Im关z兴 =  and nonreal values otherwise on the parallelogram Y. Further, 关0 , ␣兴 苹 x 哫 㜷共x + i兲 共关0 , 兴 苹 x 哫 㜷共␣ + ix兲兲 is strictly increasing 共decreasing兲. Let ei = 㜷共i兲 共i = 1 , 2 , 3兲. Clearly, e3 ⬍ e2 ⬍ e1 and 㜷共␣ − ␣0 + i兲 = 㜷共␣ + ␣0 + i兲 ¬ e23 苸 共e3,e2兲
共23兲
is a real number between e3 and e2. Let ⍀per = 兵z + : z 苸 ⍀ , 苸 L其 be the periodic extension of ⍀. Following Ref. 17, we define ⌿共z兲 = =
共20兲
We will give explicit solutions to Eqs. 共17兲 and 共19兲 for a rectangular or a rhombic unit cell using Weierstrass elliptic functions. In complex analysis, we denote by z = x1 + ix2 共x1 , x2 苸 IR兲 a point on the complex plane C. We identify the complex plane C with IR2 in this obvious manner. Let 21 , 23 苸 C with Im关1 / 3兴 ⫽ 0 be the periods and
册
is Y-periodic, analytic on Y, has a second-order pole at every lattice point in L, and takes the same value at any two points which are symmetric with respect to 2. For more detailed discussions of 㜷共z 兩 1 , 3兲, the reader is referred to the textbooks of Markushevich18 and Ahlfors.19
Since ⍀ is a slit parallel to x1-axis, the second of Eq. 共19兲 can be rewritten as
uf共x1,x2兲 + f2 = 0 x2
冋
1 1 1 2 + 兺 2 − 2 z 苸L\兵0其 共z − 兲
冕 冑
␥共0,z兲
共z1兲dz1 ¬ U共x1,x2兲 + iV共x1,x2兲,
㜷共z兲 − e2 , 㜷共z兲 − e23
共z兲 共24兲
where ␥共0 , z兲 denotes a rectifiable integration path contained ¯ , U共V兲 : IR2 \ ⍀ ¯ → IR is the real 共imaginary兲 part of in C \ ⍀ per per ⌿, and the square root takes values only from the branch with 冑1 = 1 共the branch cut is along the negative real axis兲. Grabovsky and Kohn17 showed that ⌿共z兲 is single-valued, ¯ , and hence satisfies the Cauchy–Riemann analytic on C \ ⍀ per equation
U V = , x1 x2
U V =− x2 x1
on
¯ . C\⍀ per
共25兲
Let
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t1 = ⌿共␣兲 =
冕
␥1
共z兲dz and it3 = ⌿共i兲 =
冕
␥2
共z兲dz, 共26兲
where ␥1 = 兵x1 + ix2 : 0 ⱕ x1 ⱕ ␣ , x2 = 0其 and ␥2 = 兵x1 + ix2 : 0 ⱕ x2 ⱕ  , x1 = 0其. Since IR 苹 㜷共z兲 ⱖ e1 on ␥1, IR 苹 㜷共z兲 ⱕ e3 on ␥2, and Re关共z兲兴 = 0 ∀ z 苸 ⍀, we see that t1 , t3 ⬎ 0,
U共x1,x2兲 V共x1,x2兲 = =0 x1 x2
∀ 共x1,x2兲 苸 ⍀.
共27兲 FIG. 3. 共Color online兲 Rhombic unit cell.
In particular, we notice that U is continuous on IR2, but V is discontinuous across the slit ⍀. Since d / dz关⌿共z + 2␣兲 − ⌿共z兲兴 = 0 and d / dz关⌿共z + 2i兲 − ⌿共z兲兴 = 0, by Eq. 共26兲 we have
再
⌿共z + 2␣兲 = ⌿共z兲 + 2t1 ⌿共z + 2i兲 = ⌿共z兲 + 2it3
冎
¯ . ∀z苸C\⍀ per
+
V共0,x2兲 = V共2␣,x2兲,
= k0 f 22 +
共28兲
V共x1,0兲 = V共x1,2兲 − 2t3,
苸 关0,2␣兴.
∀ x1 共29兲
␣f1 U共x1,x2兲 − f 1x1 t1
¯ 共30兲 ∀ 共x1,x2兲 苸 Y \ ⍀
is a solution of Eq. 共17兲. To show this, we notice that uf defined by Eq. 共30兲 satisfies the first and second of Eq. 共17兲 关see Eqs. 共18兲 and 共27兲兴. The last of Eq. 共17兲, i.e.,
= uf共2␣,x2兲
= uf共x1,2兲
∀ x1 苸 关0,2␣兴,
=
k0 4␣ +
冕
2
0
冤 冥
␣t3 0 1 e A = t1 . k0 0 1
It is convenient to relate the effective conductivity directly with the geometric parameters, e.g., the length of the slit 2␣0. To this end, we note that 共see Ref. 18兲
冋 册 d㜷共z兲 dz
2␣
2 f 2
0
2␣ f 1
冉
冉
冊
uf + f 2 dx1 x2
冊 册
uf + f 1 dx2 x1
k0 ␣ f 1 = k0 f 22 + 4␣ t1
冋冕
2␣
0
U 2 f 2 dx1 x2
2
= 4关㜷共z兲 − e1兴关㜷共z兲 − e2兴关㜷共z兲 − e3兴,
1
冑e1 − e3
冕
K共m兲 =
1
0
共31兲
K
冉
冊
e2 − e3 , e1 − e3
=
1
冑e1 − e3
K
冉
冊
e1 − e2 , e1 − e3
dt 共1 − t 兲共1 − mt2兲 2
is the complete elliptic integral of the first kind. Changing the integration variable from z to 㜷 = 㜷共z兲, by Eq. 共33兲 we write Eq. 共26兲 as t1 =
冕
+⬁
=
t3 = =
1
冑e1 − e3 K
冕
1
冑4共㜷 − e1兲共㜷 − e23兲共㜷 − e3兲 d㜷
e1
共f · x兲n · A共x兲共ⵜuf + f兲
共32兲
where
follows from Eq. 共29兲. We now calculate the effective conductivity tensor Ae. Note that Eq. 共6兲 is not applicable for k1 = ⬁ and = 0, and we shall use Eq. 共5兲. From Eqs. 共25兲, 共29兲, and 共30兲 we find
Y
冊
␣t3 . t1
共33兲
∀ x2 苸 关0,2兴,
f 1␣ f 1␣ U共x1,0兲 = U共x1,2兲 t1 t1
冕 冋冕
冉
2
␣=
f 1␣ f 1␣ U共0,x2兲 = U共2␣,x2兲 − 2f 1␣ uf共0,x2兲 = t1 t1
1 f·A f= 4␣
k0 ␣ f 1 =2␣ 关− 2 f 2V共x1,x2兲兩xx1=0 1 4␣ t1
∀ x2
We claim that
e
册
That is,
U共x1,0兲 = U共x1,2兲,
uf共x1,0兲 =
U dx2 x1
=2 + 2␣ f 1V共x1,x2兲兩xx2=0 兴 = k0 f 22 + f 21
苸 关0,2兴,
uf共x1,x2兲 =
2␣ f 1
0
From Eq. 共28兲 and the first of Eq. 共24兲, we obtain U共0,x2兲 = U共2␣,x2兲 − 2t1,
冕
2
冉
e3
冊
e23 − e3 , e1 − e3 1
d㜷 −⬁ 冑− 4共㜷 − e1兲共㜷 − e23兲共㜷 − e3兲 1
冑e1 − e3 K
冉
冊
e1 − e23 . e1 − e3
共34兲
Therefore,
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103503-6
J. Appl. Phys. 105, 103503 共2009兲
Liping Liu
4
3.5
3 α t3/β t1
L=2.5 2.5
L=2
2
L=1
FIG. 4. 共Color online兲 The effective conductivity in x1-direction vs the length of the slit. From up to down, the unmarked curves are calculated for rectangular unit cells with 2 = d = 2 and 2␣ = L = 2.5, 2 , 1.5, 1. The + markers are calculated for rhombic unit cells with 2 = d = 2 and 4␣ = L = 2.5, 2 , 1.5, 1 共see also Figs. 8 and 9兲.
L=1.5
1.5
1 0
0.1
0.2
0.3
0.4
0.5 α /α=l/L
0.6
0.7
0.8
0.9
1
0
␣t3 K共m0兲K共1 − m⬘兲 , = t1 K共1 − m0兲K共m⬘兲
m0 =
e2 − e3 , e1 − e3
m⬘ =
共35兲 We now discuss how the effective conductivity depends on the geometric parameters of the microstructures. First, let us fix the unit cell and calculate ␣t3 / t1 as a function of ␣0. Setting  = d / 2 and ␣ = L / 2, we have periodic slits as shown in Fig. 8. By Eqs. 共34兲 and 共35兲 we compute ␣t3 / t1 versus ␣0 苸 共0 , 1兲 shown by the unmarked curves in Fig. 4. From up to down, the curves are calculated for  = 1 and 2␣ = L = 2.5, 2 , 1.5, 1. Immediately, we see the effective conductivity along x1-direction increases from one to infinity as the length of the slit increases from 0 to 2␣. In Fig. 5 we set ␣ = 1 and compute ␣t3 / t1 versus  苸 共0 , 1兲. From up to down, the half length of the slit, ␣0, is 0.9, 0.8, and 0.4. We see that the larger  is, the smaller effect the perfectly conducting slits have on the effective conductivity. In the limit of  → + ⬁, the effective conductivity of the composite shall be the same as the matrix if ␣0 ⫽ ␣. To measure the effect of the slits in this limit, we define a dimensionless quantity
冋
册
 ␣ t 3共 ␣ ,  , ␣ 0兲 −1 . →+⬁ ␣ t1共␣, , ␣0兲
e23 − e3 . e1 − e3
␦ = lim
Physically k0␦␣2 f 21 can be interpreted as the energy of the stray field of an infinite vertical strip of width 2␣ in the presence of a periodic array of perfectly conducting slits and under the application a uniform far field f = 共f 1 , f 2兲. By dimensional analysis we infer ␦ = ␦共␣0 / ␣兲. An analytic expression of it is desirable but not obvious. We turn to numerical method. Figure 6 shows the curve ␦ = ␦共␣0 / ␣兲. Finally, we plot the local field in the unit cell for ␣ =  = 1 and ␣0 = 0.8 in Fig. 7. We remark that the field is in fact singular around the two tips of the slit as if there are static and opposite signed charges concentrated at the two tips. We now consider the case k1 = 0. Similarly, from Eqs. 共27兲 and 共29兲 we verify that uf共x1,x2兲 =
f 2 V共x1,x2兲 − f 2x2 t3
¯ 共36兲 ∀ 共x1,x2兲 苸 Y \ ⍀
satisfies all of Eq. 共19兲. Since k1 = 0 and = 0, from the last line of Eq. 共6兲 we have
4
3.5
α0/α=0.9
α t3/β t1
3 α0/α=0.8
2.5
FIG. 5. 共Color online兲 The effective conductivity in x1-direction vs the aspect ratio of the rectangular unit cell. From up to down, the curves are calculated for ␣ = 1 and ␣0 = 0.9, 0.8, 0.4.
2
1.5 α0/α=0.4
1 0
1
2
3
4
5 β/α
6
7
8
9
10
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103503-7
J. Appl. Phys. 105, 103503 共2009兲
Liping Liu 2
2.5
1.8 2
1.6 1.4
1.5 δ
1.2 1
1 0.8
0.5
0.6 0.4
0 0
0.2
0.4
α0/α
0.6
0.8
1
0
FIG. 6. 共Color online兲 The dimensionless quantity ␦ vs ␣0 / ␣.
f · A f = k0兩f兩 + k0f · e
2
W
ⵜuf .
0.2
共37兲
0
0.5
1
1.5
2
FIG. 7. 共Color online兲 The electric field in the unit cell for ␣ =  = 1 and ␣0 = 0.8.
Y\⍀
To find the unknown integral on the right hand side of Eq. 共37兲, we notice that from the Green’s theorem or the divergence theorem,
冕冋 册
iU · nds = U
Y
册 冕 冋 冕冋 册 Y\⍀
+
⍀
U U +i dx1dx2 x2 x1 iU · nds, U
共38兲
where 共ds兲2 = 共dx1兲2 + 共dx2兲2 and n is the outward normal on Y or ⍀. Since U is continuous and bounded on Y, the last term on the right hand side of Eq. 共38兲 vanishes. From Eqs. 共25兲, 共29兲, and 共38兲 we obtain
冕 冋 Y\⍀
−
册
V V dx1dx2 = +i x1 x2
冕冋 册 Y
iU · nds = 4it1 . U 共39兲
From Eqs. 共36兲 and 共37兲 we conclude that
冋 冉 冊册
f · A f = k0 兩f兩 + e
2
f 22
t1 −1 ␣t3
冤 冥
1 0 1 e t1 . ⇒ A = 0 k0 ␣t3 共40兲
We remark that Eq. 共40兲 can also be obtained from Eq. 共32兲 by the duality transformation.4,5,15
notations in this section for the same type of quantities as in the last section. Within the context this shall not give rise to confusion. From Markushevich18 we know that 㜷共z兲 takes real values on the vertical lines Re关z兴 = 2␣ and horizontal lines Im关z兴 = 0 and nonreal values otherwise on the open parallelogram Y. Let ei = 㜷共i兲. We further know that e1 =¯e3 is not a real number and 共0 , 2␣兴 苹 x 哫 㜷共x兲 strictly decreases from +⬁ to e2. Let e23 = 㜷共2␣ − ␣0兲 = 㜷共2␣ + ␣0兲 苸 共e2, + ⬁兲 and define, as in Eq. 共24兲, ⌿共z兲 = =
冕 冑
␥共0,z兲
共z1兲dz1 ¬ U共x1,x2兲 + iV共x1,x2兲,
㜷共z兲 − e2 . 㜷共z兲 − e23
In this section we assume that 1 = ␣ − i, 2 = 2␣, and 3 = ␣ + i 共␣ ,  ⬎ 0兲 and that the slit ⍀ = 兵x1 : 2␣ − ␣0 ⬍ x1 ⬍ 2␣ + ␣0其 共0 ⬍ ␣0 ⬍ 2␣兲 lies on the x1-axis 共see Fig. 3兲. In this case, Y is an open rhombus with side length of 2共␣2 + 2兲1/2 and area of 8␣. We again briefly write 㜷共z兲 = 㜷共z 兩 1 = ␣ − i , 3 = ␣ + i兲. Accordingly, we use the same
共41兲
We can similarly verify that ⌿共z兲 is single-valued, ana¯ , and hence satisfies the Cauchy–Riemann Eq. lytic on C \ ⍀ per ¯ . Let 共25兲 on C \ ⍀ per t1 − it3 = 21 ⌿共2␣ − 2i兲 =
1 2
冕
␥1
共z兲dz,
共42兲
where ␥1 is the straight path from 0 to 21 = 2␣ − 2i. Since ¯兲 and 冑z = 冑¯, 㜷共z兲 = 㜷共z z we have t1 + it3 = 21 ⌿共2␣ + 2i兲 =
B. Rhombic unit cell
共z兲
1 2
冕
␥2
共z兲dz,
共43兲
where ␥2 is the straight path from 0 to 23 = 2␣ + 2i. Let ␥3 = 兵x : 0 ⱕ x ⱕ 2␣ − ␣0其. Since Im关共z兲兴 = 0 on ␥3 and Re关共z兲兴 = 0 on ⍀, we have
U共x1,x2兲 V共x1,x2兲 = =0 x1 x2
∀ 共x1,x2兲 苸 ⍀.
共44兲
Analogous to Eqs. 共28兲–共40兲, we again find that
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103503-8
J. Appl. Phys. 105, 103503 共2009兲
Liping Liu
冕冑 冕冑 ⬁
t1 = t3
0 ⬁
0
dt
t + 2t cos ⬘ + 1 4
2
dt
,
exp共2i⬘兲 =
e23 − e1 , e23 − e3
t4 − 2t2 cos ⬘ + 1 共46兲
FIG. 8. 共Color online兲 Periodic slits arranged in a rectangular lattice.
uf共x1,x2兲 =
␣f1 U共x1,x2兲 − f 1x1 t1
¯ ∀ 共x1,x2兲 苸 Y \ ⍀
is a solution of Eq. 共17兲, and therefore,
冤 冥
␣t3 0 1 e A = t1 . k0 0 1
共45兲
The reader is invited to carry out the detailed calculations. More, from Ref. 18 we find
冕冑 冕冑 ⬁
␣ = 
0 ⬁ 0
dt
t + 2t cos 0 + 1 4
2
dt
t4 − 2t2 cos 0 + 1
,
e2 − e1 exp共2i0兲 = , e2 − e3
where 0 ⬍ 0 , ⬘ ⬍ . To investigate the effects of different lattices or unit cells, we compare the following two configurations 共see Figs. 8 and 9兲. It is not hard to see that the periodic slits in Fig. 8 correspond to a rectangular unit cell in Fig. 2 with ␣ = L / 2,  = d / 2, and ␣0 = l / 2, whereas the periodic slits in Fig. 9 correspond to a rhombic unit cell in Fig. 3 with ␣ = L / 4,  = d / 2, and ␣0 = l / 4. Meanwhile, if we shift to the left by L / 2 every other layer of the slits in Fig. 8, we obtain the configuration in Fig. 9. Using Eqs. 共35兲 and 共46兲, in Fig. 4 we plot the effective conductivities along x1-direction of these two configurations against l / L by unmarked curve and “+” markers, respectively. From up to down, the geometric parameters in Figs. 8 and 9 are chosen as d = 2, l = 1, and L = 2.5, 2 , 1.5, 1. We observe that there is no discernible difference between the unmarked curve and + markers in Fig. 4, which means the effective conductivities of the two configurations in Figs. 8 and 9 are the same, at least to the extent of our numerical resolution. The reader may speculate that this arises from symmetry; the author remarks that elementary arguments by symmetry cannot prove this unexpected result. For the case k1 = 0, by similar arguments we verify that uf共x1,x2兲 =
f 2 V共x1,x2兲 − f 2x2 t3
¯ 共47兲 ∀ 共x1,x2兲 苸 Y \ ⍀
satisfies all of Eq. 共19兲. Parallel to Eqs. 共37兲–共40兲 or simply by the duality transformation, we can show the effective con-
FIG. 9. 共Color online兲 Periodic slits arranged in a rhombic lattice.
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103503-9
J. Appl. Phys. 105, 103503 共2009兲
Liping Liu
ductivity is given by Eq. 共40兲 with ␣, , t1, and t3 interpreted as in Eq. 共45兲. V. SUMMARY AND DISCUSSION
We derive a closed-form formula for the effective conductivity of composites with periodic E-inclusion microstructure. When the periodic E-inclusion degenerates to a periodic array of slits, we give explicit solutions to local fields and the effective conductivity of the composite with singular inhomogeneities. Through a linear transformation, these results can be extended to two-phase composites of any anisotropic materials. The results of this paper can be used in the following ways. In the first place the closed-form formula 共13兲 with the volume fraction and the shape matrix Q as parameters can be used to give a quick estimate of the effective properties of a composite. In reality, of course, it is questionable to what extent the microstructure of a composite can be approximated by a periodic E-inclusion. However our prediction 关see Eq. 共13兲兴 is at least physical and realizable, and the qualitative feature of how the effective properties depend on the volume fraction and shape of the inhomogeneities should remain regardless of the exact shapes of the inhomogeneities. Second, the analytic results provide a benchmark for testing various empirical models and numerical codes. Last but not least important, the results in Sec. IV, in particular the comparison between rectangular and rhombic unit cells 共see Fig. 4兲, suggest that the effective properties of composites with a singular phase are predominantly determined by the distance
between nearby inhomogeneities. This has been observed by various authors and is in fact the basis of the network model.6–9 What is noteworthy here is that this remains to be true even for an extreme shape such as a slit, for which the field is not localized between slits 共see Fig. 7兲. D. J. Bergman, Phys. Rep., Phys. Lett. 43, 377 共1978兲. R. Hill, J. Mech. Phys. Solids 13, 213 共1965兲. 3 Z. Hashin and S. Shtrikman, J. Appl. Phys. 33, 3125 共1962兲. 4 J. B. Keller, J. Appl. Phys. 34, 991 共1963兲. 5 A. Dykne, Sov. Phys. JETP 32, 63 共1971兲. 6 V. Ambegaokar, B. I. Halperin, and J. S. Langer, Phys. Rev. B 4, 2612 共1971兲. 7 J. P. Clerc, G. Giraud, J. M. Laugier, and J. M. Luck, Adv. Phys. 39, 191–309 共1990兲. 8 S. M. Kozlov, Russ. Math. Surv. 44, 91 共1989兲. 9 L. Berlyand and A. Kolpakov, Arch. Ration. Mech. Anal. 159, 179 共2001兲. 10 W. R. Delameter, G. Herrmann, and D. M. Barnett, ASME Trans. J. Appl. Mech. 42, 74 共1975兲. 11 J. Wang, J. Fang, and B. L. Karihaloo, Int. J. Solids Struct. 37, 4261 共2000兲. 12 S. B. Vigdergauz, Inzh. Zh., Mekh. Tverd. Tela 21, 165–169 共1986兲. 13 L. P. Liu, R. D. James, and P. H. Leo, Arch. Ration. Mech. Anal. 共unpublished兲. 14 A. Friedman, Variational Principles and Free Boundary Problems 共Wiley, New York, 1982兲. 15 G. W. Milton, The Theory of Composites 共Cambridge University Press, Cambridge, 2002兲. 16 L. P. Liu, R. D. James, and P. H. Leo, Metall. Mater. Trans. A 38, 781 共2007兲. 17 Y. Grabovsky and R. V. Kohn, J. Mech. Phys. Solids 43, 949 共1995兲. 18 A. I. Markushevich, Theory of Function of Complex Variable 共Chelsea, New York, 1977兲, Vol. 1–3. 19 L. V. Ahlfors, Complex Analysis 共McGraw-Hill, New York, 1979兲. 1 2
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Effective conductivities of two-phase composites with a singular phase Liping Liua兲 Department of Mechanical Engineering, University of Houston, Houston, Texas 77204, USA
共Received 17 January 2009; accepted 3 March 2009; published online 18 May 2009兲 We calculate the effective conductivity of a two-phase composite with a periodic array of inhomogeneities. The shape of the inhomogeneities is assumed to be a periodic E-inclusion. The effective conductivity is expressed in terms of the volume fraction of the inhomogeneities and a matrix, which characterizes the shape of the periodic E-inclusion. This solution is rigorous, closed-form, and applicable to situations that the conductivity of the inhomogeneities is singular, i.e., zero or infinite. Further, when the periodic E-inclusion degenerates to a periodic array of slits with vanishing volume fraction, we give explicit solutions to local fields and effective conductivity of the composite with singular inhomogeneities. © 2009 American Institute of Physics. 关DOI: 10.1063/1.3110026兴 I. INTRODUCTION
High-contrast composites have wide industrial applications. Examples include metal and glass foams, fibrous metal and glass materials, fiber-reinforced materials, and fractured porous media. Once the notion of effective properties of composites or heterogeneous media is established, we are faced with the critical problem of finding the effective properties of a given composite and, if possible, relating the effective properties with the microstructure of the composite. However this is difficult even for composites with simple microstructures, for example, a two-phase composite with a periodic array of inhomogeneous spheres. In this and similar cases, heuristic methods such as the effective medium theory 共Ref. 1 and references therein兲 and the self-consistent method2 can give us closed-form formulas of the effective properties. These approximate formulas are in good agreement with experiments when the contrast between the two phases is small but unsatisfactory when the contrast of the constituent phases becomes large. If we idealize the physical properties of the inhomogeneities to be infinite or zero, many of the approximate formulas even yield unphysical results. Further, rigorous bounds such as the Hashin–Shtrikman bounds3 are not of much use since they are far apart from each other for high-contrast composites. So it is important for both theory and application to have a reasonable estimate of the effective properties of high-contrast composites. This is the purpose of this paper. Theoretical results are available for a small number of particular cases in the literature. Keller4 considered a composite of a cubic array of perfectly conducting spheres embedded in a normal conducting medium and obtained an asymptotic formula for the effective conductivity when the gaps between nearby spheres are small. Dykne5 showed that a two-phase system of insulating and conducting materials could undergo a dielectrics-conductor transition at certain critical concentration. For random high-contrast composites, a number of authors6,7 used a discrete network to model a high-contrast composite based on physical consideration. a兲
Electronic mail: [email protected].
0021-8979/2009/105共10兲/103503/9/$25.00
Kozlov8 and Berlyand and Kolpakov9 showed that the discrete network is a sound model of the original continuum problem. From these works, we observe that high-contrast composites have two distinguishing features compared with normal composites. The first is that high-contrast composites can undergo a percolation transition, and the second is that the effective properties are dependent on the local geometry of the composite, say, the interinhomogeneity distance, as much as the global parameter, such as the volume fraction. In particular, if the physical property of the inhomogeneities is zero or infinite, the inhomogeneities could have a significant impact on the effective properties even if their volume fraction vanishes. This is well understood in the context of fracture mechanics10,11 but seems unnoticed for conductive composites. In this paper we give explicit solutions to the effective properties of a class of periodic composites. To obtain these solutions we choose a special class of microstructures called periodic E-inclusions or Vigdergauz12 structures in two dimensions.13 Unlike previous examples, our solutions are rigorous and closed-form, which express the effective properties in terms of the volume fraction of the inhomogeneities and the matrix Q, which characterizes the shape of the inhomogeneities. A disadvantage of our solutions is that the shape of periodic E-inclusions is not directly given; we need to solve a variational inequality to find a periodic E-inclusion.13,14 Nevertheless, without solving the variational inequality we know qualitatively the shapes of periodic E-inclusions and how they depend on the volume fraction and the shape matrix Q. In particular, when the shape matrix Q is singular and the volume fraction approaches to zero, the periodic E-inclusion degenerates to a slit in two dimensions. In this case we express the effective properties in terms of the length of the slit. It indicates that the widely used rules of mixtures, which interpolate between the properties of the matrix and the inhomogeneities by volume fractions, could be qualitatively misleading for high-contrast composites. Two remarks are in order here about the scope of this paper. For simplicity we discuss the conductivity problem,
105, 103503-1
© 2009 American Institute of Physics
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103503-2
J. Appl. Phys. 105, 103503 共2009兲
Liping Liu
but note that the solutions are applicable to other physical properties including dielectric properties, diffusive properties, transport properties, and at least qualitatively, elastic properties. More, our solutions are for periodic composites. Therefore, the effects of percolation and randomness are not addressed here. The paper is organized as follows. In Sec. II we formulate the governing equation for a periodic composite and derive the formulas for calculating the effective properties of the composite. In Sec. III we define the periodic E-inclusion and show how to calculate the effective properties of a composite with a periodic E-inclusion microstructure. In Sec. IV we discuss a particular situation that the periodic E-inclusion degenerates to a periodic array of slits. For this particular situation we give explicit solutions to local fields and effective properties for a rectangular lattice in Sec. IV A and for a rhombic lattice in Sec. IV B. Finally we summarize our results in Sec. V. II. PROBLEM FORMULATION
Let Y 傺 IRn be a unit cell associated with a Bravais lattice L, ⍀ 傺 Y be an inclusion containing an inhomogeneity, and be the volume fraction of the inhomogeneities. Consider a periodic two-phase composite with conductivity given by A共x兲 =
再
k0I if x 苸 Y \ ⍀ k1I if x 苸 ⍀,
冎
共1兲
where k0 ⬎ 0, k1 ⱖ 0, and I 苸 IRn⫻n is the identity matrix. For ease of terminology, we refer to phase-0 as the matrix and phase-1 as the inhomogeneities. From the homogenization theory,15 the effective conductivity of the composite, described by a symmetric tensor Ae, is given by f · Aef = min u苸W
W
共ⵜu + f兲 · A共x兲共ⵜu + f兲,
共2兲
Y
where WV = 共1 / 兩V兩兲兰V denotes the average of the integrand over V 共兩V兩 denotes the volume of V兲, f 苸 IRn is the average applied field, and the admissible space W is the collection of all periodic square integrable functions u : IRn → IR whose gradients remain square integrable. To evaluate the effective conductivity tensor Ae, we need to solve the Euler–Lagrange equation of Eq. 共2兲 for the minimizer uf,
再
div关A共x兲共ⵜuf + f兲兴 = 0 on Y , periodic boundary conditions on Y ,
冎
共3兲
and then compute the integral for u = uf on the right hand side of Eq. 共2兲. The effective conductivity of a composite can be expressed as a boundary integral of the minimizer uf or an integral on one of the phases. To see this, we notice uf being a periodic function and A共x兲共ⵜuf + f兲 being divergence-free imply
冕
ⵜuf = 0
and
Y
Therefore, we have
冕
Y
ⵜuf · A共x兲共ⵜuf + f兲 = 0.
共4兲
f · A ef =
W W
共ⵜuf + f兲 · A共x兲共ⵜuf + f兲
Y
=
f · A共x兲共ⵜuf + f兲 = f ·
Y
=
1 兩Y兩
冕
Y
W
共ⵜx兲A共x兲共ⵜuf + f兲
Y
共f · x兲n · A共x兲共ⵜuf + f兲,
共5兲
where n is the outward normal on Y. Alternatively, from the first line in Eq. 共5兲 we find f · A ef = k 0
W
W W
f · 共ⵜuf + f兲 + 共k1 − k0兲f ·
Y
冋
= k0兩f兩2 + 共k1 − k0兲f · f − 共1 − 兲
⍀
共ⵜuf + f兲
Y\⍀
册
共ⵜuf + f兲 . 共6兲
For general inclusions, we do not have a closed-form solution of Eq. 共2兲, and thus much of the works have been focused on the bounds on the effective conductivity tensor Ae and numerical methods that compute Ae for a given inclusion ⍀. An exception is the case that the inclusion ⍀ is a periodic E-inclusion. Below we describe what a periodic E-inclusion is and calculate the effective conductivity of a composite with the inclusion being a periodic E-inclusion.
III. EFFECTIVE CONDUCTIVITIES OF COMPOSITES WITH PERIODIC E-INCLUSION INHOMOGENEITIES
Recently, the Liu et al.16 found a class of special inclusions for which a closed-form solution of Eq. 共2兲 is available. These special inclusions, termed as periodic E-inclusions, are defined as an inclusion ⍀ 傺 Y such that the overdetermined problem
冦
⌬ = − ⍀ on Y , ⵜⵜ = − 共1 − 兲Q on ⍀, periodic boundary conditions on Y
冧
共7兲
admits a solution where ⍀ is the characteristic function of ⍀ and Q 苸 Q ª 兵M n⫻n :M is positive semidefinite with Tr共M兲 苸 IRsym
= 1其.
共8兲
The Liu et al.13 showed the existence of periodic E-inclusions for any volume fraction 苸 共0 , 1兲, any matrix Q 苸 Q, and any Bravais lattice L. Figure 1 shows three examples of periodic E-inclusions in two dimensions 共see Refs. 17, 16, and 13 for more examples兲. We remark that periodic E-inclusions are generalizations of two-dimensional structures constructed by Vigdergauz.12 If the inclusion ⍀ for the composite is a periodic E-inclusion with matrix Q and volume fraction , we claim that a solution of Eq. 共3兲 is given by
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103503-3
J. Appl. Phys. 105, 103503 共2009兲
Liping Liu
1
Ae = k0I + 共k1 − k0兲I − 共1 − 兲共k1 − k0兲2Q关共1 − 兲共k1 − k0兲Q + k0I兴−1 .
0.8
共13兲
We remark that the above formula is rigorous and attains the lower 共upper兲 Hashin–Shtrikman3 bound if k0 ⬍ k1 共k0 ⬎ k1兲. It has a few regimes that need separate attention.
0.6 0.4
共1兲 If k1 is nonsingular, i.e., ⫽0 or +⬁, formula 共13兲 gives a definite answer to the effective conductivity tensor of the composite of a periodic E-inclusion with any matrix Q 苸 Q and any volume fraction 苸 关0 , 1兴 since 共1 − 兲 ⫻共k1 − k0兲Q + k0I is invertible. In particular, if we assume the composite is isotropic, then the corresponding matrix Q is equal to I / n. Denoting by ke the isotropic conductivity of the composite, from Eq. 共13兲 we have
0.2 0 −0.2 −0.4 −0.6
ke 共1 − 兲共n − 1兲 + 关1 + 共n − 1兲兴k1/k0 = . k0 n − 1 + + 共1 − 兲k1/k0
−0.8 −1 −1
−0.5
0
0.5
1
FIG. 1. 共Color online兲 From outward to inward, the regions bounded by the curves are periodic E-inclusions with matrices Q and volume fractions given by Eq. 共16兲 and unit cell Y = 共−1 , 1兲2.
共2兲 If Q is positive definite, formula 共13兲 also gives a definite answer even if k1 is singular. In particular, we have Ae/k0 = 共1 − 兲I − 共1 − 兲Q关− 共1 − 兲Q + I兴−1 if k1 = 0, Ae/k0 = I +
uf = a · ⵜ,
a = 共k1 − k0兲关共1 − 兲共k1 − k0兲Q + k0I兴−1f. 共9兲
To see this, we notice that for Eq. 共3兲 the interfacial condition on ⍀ can be written as k1n · 关ⵜuf共x−兲 + f兴 = k0n · 关ⵜuf共x+兲 + f兴
∀ x 苸 ⍀, 共10兲
whereas being a solution of ⌬ = − ⍀ satisfies ⵜⵜ共x−兲 − ⵜⵜ共x+兲 = − n 丢 n
∀ x 苸 ⍀,
冋
共1 − 兲Q + =
= k1n · 关− 共1 − 兲Qa + f兴 − k1关− 共1 − 兲n · Qa + n · f兴 + k0关− 共1 − 兲n · Qa + a · n + n · f兴 = k1n · 关ⵜa · ⵜ共x−兲 + f兴 + n · 关共k1 − k0兲共1 − 兲Qa + k0a + 共k0 − k1兲f兴. 共12兲 Therefore, if we choose the vector a as in Eq. 共9兲, the last term on the right hand side of Eq. 共12兲 vanishes, and hence uf = a · ⵜ satisfies the interfacial condition 共10兲. Further, we can easily verify that uf = a · ⵜ satisfies the first of Eq. 共3兲 on the interior and exterior of ⍀ and the periodic boundary conditions on Y. We thus conclude that uf = a · ⵜ is a solution of Eq. 共3兲 if ⍀ is a periodic E-inclusion and is given by Eq. 共7兲. Using Eq. 共9兲 we can calculate the effective conductivity tensor Ae for a periodic E-inclusion. By the first line of Eq. 共6兲 and the second of Eq. 共7兲 we find
共15兲
k0 I k1 − k0
册
−1
1 k0 Q−1Q−1 Q−1 − 2 1− 共1 − 兲 共k1 − k0兲
共11兲
k0n · 关ⵜa · ⵜ共x+兲 + f兴
Q−1 if k1 = + ⬁. 1−
The second of the above formula follows from the fact that as k1 → + ⬁,
+O
where n is the outward normal on ⍀ and x− 共x + 兲 denotes the limit from the inside 共outside兲 of ⍀. From Eq. 共11兲 and the second of Eq. 共7兲, direct calculation reveals that for any a 苸 IRn,
共14兲
冉冏
k0 k1 − k0
冏冊 2
.
If the composite is assumed to be isotropic with the effective conductivity denoted by ke0 共k⬁e 兲 for k1 = 0 共k1 = + ⬁兲, then from Eq. 共14兲 or Eq. 共15兲 we have ke0 共1 − 兲共n − 1兲 = , k0 n−1+
k⬁e 1 + 共n − 1兲 = . k0 1−
共3兲 If 苸 共0 , 1兴, Q is singular, and k1 = 0 or +⬁, formula 共15兲 yields a definite answer to Ae if we interpret the inverse of a matrix M 苸 IRn⫻n as M−1 = lim 关M + I兴−1 . 0
共4兲 If = 0 and Q is singular, formula 共15兲 is not necessarily meaningful since we could get a term such as 0 · ⬁. Nevertheless, such situations are physically interesting. Below we focus on this regime in two dimensions.
IV. EFFECTIVE CONDUCTIVITIES OF COMPOSITES WITH PERIODIC SLIT INHOMOGENEITIES
In this section we restrict ourselves to two dimensions 共n = 2兲. From the viewpoint of last section and in connection with geometry, we are interested in the following limit: we
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103503-4
J. Appl. Phys. 105, 103503 共2009兲
Liping Liu
fix the two ends of the periodic E-inclusion ⍀ along x1-direction and let the dimension of ⍀ along x2-direction shrink to zero. Using formulas in Refs. 17 and 16, we plot three periodic E-inclusions in two dimensions for unit cell Y = 共−1 , 1兲2 in Fig. 1. From outward to inward, the matrix Q and volume fraction are given by
Q=
冤
r 1+r
0
0
1 1+r
冥
,
r = 0.5, 0.2, 0.01
and
FIG. 2. 共Color online兲 Rectangular unit cell.
共16兲
= 0.42, 0.17, 0.04. We note that as Q→
冋 册 0 0 0 1
and
㜷共z兩1, 3兲 =
→ 0,
the periodic E-inclusion approaches to a slit parallel to x1-axis. Our goal is to compute the effective conductivity Ae in this limit for singular k1, for which Eq. 共13兲 or Eq. 共15兲 is no longer applicable. If k1 is a finite positive number, from Eq. 共13兲 we have Ae = k0I, as expected. To proceed, we reformulate problem 共3兲 for k1 = + ⬁ or 0. If k1 = + ⬁, the inclusion ⍀ necessarily remains as an equipotential body, and so problem 共3兲 is equivalent to
冦
¯, ⌬uf = 0 on Y \ ⍀ t · 共ⵜuf + f兲 = 0 on ⍀, periodic boundary conditions on Y ,
冧
共17兲
where t is the tangent on ⍀. Since ⍀ is a slit parallel to x1-axis, the second of Eq. 共17兲 can be rewritten as
uf共x1,x2兲 + f1 = 0 x1
∀ 共x1,x2兲 苸 ⍀.
共18兲
If k1 = 0, from the Gaussian theorem we infer that problem 共3兲 is equivalent to
冦
¯, ⌬uf = 0 on Y \ ⍀ n · 共ⵜuf + f兲 = 0 on ⍀, periodic boundary conditions on Y .
冧
共19兲
∀ 共x1,x2兲 苸 ⍀.
L = 兵211 + 223:1, 2 苸 Z其
共21兲
be the lattice. Associated with this lattice, we denote by Y the open parallelogram with vertices 0, 21, 22 = 2共1 + 3兲, and 23 and recall that the Weierstrass 㜷-function
共22兲
A. Rectangular unit cell
We first assume that 1 = ␣, 3 = i 共␣ ,  ⬎ 0兲 and that the slit ⍀ = 兵x1 + i : ␣ − ␣0 ⬍ x1 ⬍ ␣ + ␣0其 共0 ⬍ ␣0 ⬍ ␣兲 lies on the horizontal line Im关z兴 = . In this case, Y is an open rectangular with base 2␣ and height 2 共see Fig. 2兲. We briefly write 㜷共z兲 = 㜷共z 兩 1 = ␣ , 3 = i兲 in this section. From Markushevich18 we know that 㜷共z兲 takes real values on the vertical lines Re关z兴 = ␣ and horizontal lines Im关z兴 =  and nonreal values otherwise on the parallelogram Y. Further, 关0 , ␣兴 苹 x 哫 㜷共x + i兲 共关0 , 兴 苹 x 哫 㜷共␣ + ix兲兲 is strictly increasing 共decreasing兲. Let ei = 㜷共i兲 共i = 1 , 2 , 3兲. Clearly, e3 ⬍ e2 ⬍ e1 and 㜷共␣ − ␣0 + i兲 = 㜷共␣ + ␣0 + i兲 ¬ e23 苸 共e3,e2兲
共23兲
is a real number between e3 and e2. Let ⍀per = 兵z + : z 苸 ⍀ , 苸 L其 be the periodic extension of ⍀. Following Ref. 17, we define ⌿共z兲 = =
共20兲
We will give explicit solutions to Eqs. 共17兲 and 共19兲 for a rectangular or a rhombic unit cell using Weierstrass elliptic functions. In complex analysis, we denote by z = x1 + ix2 共x1 , x2 苸 IR兲 a point on the complex plane C. We identify the complex plane C with IR2 in this obvious manner. Let 21 , 23 苸 C with Im关1 / 3兴 ⫽ 0 be the periods and
册
is Y-periodic, analytic on Y, has a second-order pole at every lattice point in L, and takes the same value at any two points which are symmetric with respect to 2. For more detailed discussions of 㜷共z 兩 1 , 3兲, the reader is referred to the textbooks of Markushevich18 and Ahlfors.19
Since ⍀ is a slit parallel to x1-axis, the second of Eq. 共19兲 can be rewritten as
uf共x1,x2兲 + f2 = 0 x2
冋
1 1 1 2 + 兺 2 − 2 z 苸L\兵0其 共z − 兲
冕 冑
␥共0,z兲
共z1兲dz1 ¬ U共x1,x2兲 + iV共x1,x2兲,
㜷共z兲 − e2 , 㜷共z兲 − e23
共z兲 共24兲
where ␥共0 , z兲 denotes a rectifiable integration path contained ¯ , U共V兲 : IR2 \ ⍀ ¯ → IR is the real 共imaginary兲 part of in C \ ⍀ per per ⌿, and the square root takes values only from the branch with 冑1 = 1 共the branch cut is along the negative real axis兲. Grabovsky and Kohn17 showed that ⌿共z兲 is single-valued, ¯ , and hence satisfies the Cauchy–Riemann analytic on C \ ⍀ per equation
U V = , x1 x2
U V =− x2 x1
on
¯ . C\⍀ per
共25兲
Let
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103503-5
J. Appl. Phys. 105, 103503 共2009兲
Liping Liu
t1 = ⌿共␣兲 =
冕
␥1
共z兲dz and it3 = ⌿共i兲 =
冕
␥2
共z兲dz, 共26兲
where ␥1 = 兵x1 + ix2 : 0 ⱕ x1 ⱕ ␣ , x2 = 0其 and ␥2 = 兵x1 + ix2 : 0 ⱕ x2 ⱕ  , x1 = 0其. Since IR 苹 㜷共z兲 ⱖ e1 on ␥1, IR 苹 㜷共z兲 ⱕ e3 on ␥2, and Re关共z兲兴 = 0 ∀ z 苸 ⍀, we see that t1 , t3 ⬎ 0,
U共x1,x2兲 V共x1,x2兲 = =0 x1 x2
∀ 共x1,x2兲 苸 ⍀.
共27兲 FIG. 3. 共Color online兲 Rhombic unit cell.
In particular, we notice that U is continuous on IR2, but V is discontinuous across the slit ⍀. Since d / dz关⌿共z + 2␣兲 − ⌿共z兲兴 = 0 and d / dz关⌿共z + 2i兲 − ⌿共z兲兴 = 0, by Eq. 共26兲 we have
再
⌿共z + 2␣兲 = ⌿共z兲 + 2t1 ⌿共z + 2i兲 = ⌿共z兲 + 2it3
冎
¯ . ∀z苸C\⍀ per
+
V共0,x2兲 = V共2␣,x2兲,
= k0 f 22 +
共28兲
V共x1,0兲 = V共x1,2兲 − 2t3,
苸 关0,2␣兴.
∀ x1 共29兲
␣f1 U共x1,x2兲 − f 1x1 t1
¯ 共30兲 ∀ 共x1,x2兲 苸 Y \ ⍀
is a solution of Eq. 共17兲. To show this, we notice that uf defined by Eq. 共30兲 satisfies the first and second of Eq. 共17兲 关see Eqs. 共18兲 and 共27兲兴. The last of Eq. 共17兲, i.e.,
= uf共2␣,x2兲
= uf共x1,2兲
∀ x1 苸 关0,2␣兴,
=
k0 4␣ +
冕
2
0
冤 冥
␣t3 0 1 e A = t1 . k0 0 1
It is convenient to relate the effective conductivity directly with the geometric parameters, e.g., the length of the slit 2␣0. To this end, we note that 共see Ref. 18兲
冋 册 d㜷共z兲 dz
2␣
2 f 2
0
2␣ f 1
冉
冉
冊
uf + f 2 dx1 x2
冊 册
uf + f 1 dx2 x1
k0 ␣ f 1 = k0 f 22 + 4␣ t1
冋冕
2␣
0
U 2 f 2 dx1 x2
2
= 4关㜷共z兲 − e1兴关㜷共z兲 − e2兴关㜷共z兲 − e3兴,
1
冑e1 − e3
冕
K共m兲 =
1
0
共31兲
K
冉
冊
e2 − e3 , e1 − e3
=
1
冑e1 − e3
K
冉
冊
e1 − e2 , e1 − e3
dt 共1 − t 兲共1 − mt2兲 2
is the complete elliptic integral of the first kind. Changing the integration variable from z to 㜷 = 㜷共z兲, by Eq. 共33兲 we write Eq. 共26兲 as t1 =
冕
+⬁
=
t3 = =
1
冑e1 − e3 K
冕
1
冑4共㜷 − e1兲共㜷 − e23兲共㜷 − e3兲 d㜷
e1
共f · x兲n · A共x兲共ⵜuf + f兲
共32兲
where
follows from Eq. 共29兲. We now calculate the effective conductivity tensor Ae. Note that Eq. 共6兲 is not applicable for k1 = ⬁ and = 0, and we shall use Eq. 共5兲. From Eqs. 共25兲, 共29兲, and 共30兲 we find
Y
冊
␣t3 . t1
共33兲
∀ x2 苸 关0,2兴,
f 1␣ f 1␣ U共x1,0兲 = U共x1,2兲 t1 t1
冕 冋冕
冉
2
␣=
f 1␣ f 1␣ U共0,x2兲 = U共2␣,x2兲 − 2f 1␣ uf共0,x2兲 = t1 t1
1 f·A f= 4␣
k0 ␣ f 1 =2␣ 关− 2 f 2V共x1,x2兲兩xx1=0 1 4␣ t1
∀ x2
We claim that
e
册
That is,
U共x1,0兲 = U共x1,2兲,
uf共x1,0兲 =
U dx2 x1
=2 + 2␣ f 1V共x1,x2兲兩xx2=0 兴 = k0 f 22 + f 21
苸 关0,2兴,
uf共x1,x2兲 =
2␣ f 1
0
From Eq. 共28兲 and the first of Eq. 共24兲, we obtain U共0,x2兲 = U共2␣,x2兲 − 2t1,
冕
2
冉
e3
冊
e23 − e3 , e1 − e3 1
d㜷 −⬁ 冑− 4共㜷 − e1兲共㜷 − e23兲共㜷 − e3兲 1
冑e1 − e3 K
冉
冊
e1 − e23 . e1 − e3
共34兲
Therefore,
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103503-6
J. Appl. Phys. 105, 103503 共2009兲
Liping Liu
4
3.5
3 α t3/β t1
L=2.5 2.5
L=2
2
L=1
FIG. 4. 共Color online兲 The effective conductivity in x1-direction vs the length of the slit. From up to down, the unmarked curves are calculated for rectangular unit cells with 2 = d = 2 and 2␣ = L = 2.5, 2 , 1.5, 1. The + markers are calculated for rhombic unit cells with 2 = d = 2 and 4␣ = L = 2.5, 2 , 1.5, 1 共see also Figs. 8 and 9兲.
L=1.5
1.5
1 0
0.1
0.2
0.3
0.4
0.5 α /α=l/L
0.6
0.7
0.8
0.9
1
0
␣t3 K共m0兲K共1 − m⬘兲 , = t1 K共1 − m0兲K共m⬘兲
m0 =
e2 − e3 , e1 − e3
m⬘ =
共35兲 We now discuss how the effective conductivity depends on the geometric parameters of the microstructures. First, let us fix the unit cell and calculate ␣t3 / t1 as a function of ␣0. Setting  = d / 2 and ␣ = L / 2, we have periodic slits as shown in Fig. 8. By Eqs. 共34兲 and 共35兲 we compute ␣t3 / t1 versus ␣0 苸 共0 , 1兲 shown by the unmarked curves in Fig. 4. From up to down, the curves are calculated for  = 1 and 2␣ = L = 2.5, 2 , 1.5, 1. Immediately, we see the effective conductivity along x1-direction increases from one to infinity as the length of the slit increases from 0 to 2␣. In Fig. 5 we set ␣ = 1 and compute ␣t3 / t1 versus  苸 共0 , 1兲. From up to down, the half length of the slit, ␣0, is 0.9, 0.8, and 0.4. We see that the larger  is, the smaller effect the perfectly conducting slits have on the effective conductivity. In the limit of  → + ⬁, the effective conductivity of the composite shall be the same as the matrix if ␣0 ⫽ ␣. To measure the effect of the slits in this limit, we define a dimensionless quantity
冋
册
 ␣ t 3共 ␣ ,  , ␣ 0兲 −1 . →+⬁ ␣ t1共␣, , ␣0兲
e23 − e3 . e1 − e3
␦ = lim
Physically k0␦␣2 f 21 can be interpreted as the energy of the stray field of an infinite vertical strip of width 2␣ in the presence of a periodic array of perfectly conducting slits and under the application a uniform far field f = 共f 1 , f 2兲. By dimensional analysis we infer ␦ = ␦共␣0 / ␣兲. An analytic expression of it is desirable but not obvious. We turn to numerical method. Figure 6 shows the curve ␦ = ␦共␣0 / ␣兲. Finally, we plot the local field in the unit cell for ␣ =  = 1 and ␣0 = 0.8 in Fig. 7. We remark that the field is in fact singular around the two tips of the slit as if there are static and opposite signed charges concentrated at the two tips. We now consider the case k1 = 0. Similarly, from Eqs. 共27兲 and 共29兲 we verify that uf共x1,x2兲 =
f 2 V共x1,x2兲 − f 2x2 t3
¯ 共36兲 ∀ 共x1,x2兲 苸 Y \ ⍀
satisfies all of Eq. 共19兲. Since k1 = 0 and = 0, from the last line of Eq. 共6兲 we have
4
3.5
α0/α=0.9
α t3/β t1
3 α0/α=0.8
2.5
FIG. 5. 共Color online兲 The effective conductivity in x1-direction vs the aspect ratio of the rectangular unit cell. From up to down, the curves are calculated for ␣ = 1 and ␣0 = 0.9, 0.8, 0.4.
2
1.5 α0/α=0.4
1 0
1
2
3
4
5 β/α
6
7
8
9
10
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103503-7
J. Appl. Phys. 105, 103503 共2009兲
Liping Liu 2
2.5
1.8 2
1.6 1.4
1.5 δ
1.2 1
1 0.8
0.5
0.6 0.4
0 0
0.2
0.4
α0/α
0.6
0.8
1
0
FIG. 6. 共Color online兲 The dimensionless quantity ␦ vs ␣0 / ␣.
f · A f = k0兩f兩 + k0f · e
2
W
ⵜuf .
0.2
共37兲
0
0.5
1
1.5
2
FIG. 7. 共Color online兲 The electric field in the unit cell for ␣ =  = 1 and ␣0 = 0.8.
Y\⍀
To find the unknown integral on the right hand side of Eq. 共37兲, we notice that from the Green’s theorem or the divergence theorem,
冕冋 册
iU · nds = U
Y
册 冕 冋 冕冋 册 Y\⍀
+
⍀
U U +i dx1dx2 x2 x1 iU · nds, U
共38兲
where 共ds兲2 = 共dx1兲2 + 共dx2兲2 and n is the outward normal on Y or ⍀. Since U is continuous and bounded on Y, the last term on the right hand side of Eq. 共38兲 vanishes. From Eqs. 共25兲, 共29兲, and 共38兲 we obtain
冕 冋 Y\⍀
−
册
V V dx1dx2 = +i x1 x2
冕冋 册 Y
iU · nds = 4it1 . U 共39兲
From Eqs. 共36兲 and 共37兲 we conclude that
冋 冉 冊册
f · A f = k0 兩f兩 + e
2
f 22
t1 −1 ␣t3
冤 冥
1 0 1 e t1 . ⇒ A = 0 k0 ␣t3 共40兲
We remark that Eq. 共40兲 can also be obtained from Eq. 共32兲 by the duality transformation.4,5,15
notations in this section for the same type of quantities as in the last section. Within the context this shall not give rise to confusion. From Markushevich18 we know that 㜷共z兲 takes real values on the vertical lines Re关z兴 = 2␣ and horizontal lines Im关z兴 = 0 and nonreal values otherwise on the open parallelogram Y. Let ei = 㜷共i兲. We further know that e1 =¯e3 is not a real number and 共0 , 2␣兴 苹 x 哫 㜷共x兲 strictly decreases from +⬁ to e2. Let e23 = 㜷共2␣ − ␣0兲 = 㜷共2␣ + ␣0兲 苸 共e2, + ⬁兲 and define, as in Eq. 共24兲, ⌿共z兲 = =
冕 冑
␥共0,z兲
共z1兲dz1 ¬ U共x1,x2兲 + iV共x1,x2兲,
㜷共z兲 − e2 . 㜷共z兲 − e23
In this section we assume that 1 = ␣ − i, 2 = 2␣, and 3 = ␣ + i 共␣ ,  ⬎ 0兲 and that the slit ⍀ = 兵x1 : 2␣ − ␣0 ⬍ x1 ⬍ 2␣ + ␣0其 共0 ⬍ ␣0 ⬍ 2␣兲 lies on the x1-axis 共see Fig. 3兲. In this case, Y is an open rhombus with side length of 2共␣2 + 2兲1/2 and area of 8␣. We again briefly write 㜷共z兲 = 㜷共z 兩 1 = ␣ − i , 3 = ␣ + i兲. Accordingly, we use the same
共41兲
We can similarly verify that ⌿共z兲 is single-valued, ana¯ , and hence satisfies the Cauchy–Riemann Eq. lytic on C \ ⍀ per ¯ . Let 共25兲 on C \ ⍀ per t1 − it3 = 21 ⌿共2␣ − 2i兲 =
1 2
冕
␥1
共z兲dz,
共42兲
where ␥1 is the straight path from 0 to 21 = 2␣ − 2i. Since ¯兲 and 冑z = 冑¯, 㜷共z兲 = 㜷共z z we have t1 + it3 = 21 ⌿共2␣ + 2i兲 =
B. Rhombic unit cell
共z兲
1 2
冕
␥2
共z兲dz,
共43兲
where ␥2 is the straight path from 0 to 23 = 2␣ + 2i. Let ␥3 = 兵x : 0 ⱕ x ⱕ 2␣ − ␣0其. Since Im关共z兲兴 = 0 on ␥3 and Re关共z兲兴 = 0 on ⍀, we have
U共x1,x2兲 V共x1,x2兲 = =0 x1 x2
∀ 共x1,x2兲 苸 ⍀.
共44兲
Analogous to Eqs. 共28兲–共40兲, we again find that
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103503-8
J. Appl. Phys. 105, 103503 共2009兲
Liping Liu
冕冑 冕冑 ⬁
t1 = t3
0 ⬁
0
dt
t + 2t cos ⬘ + 1 4
2
dt
,
exp共2i⬘兲 =
e23 − e1 , e23 − e3
t4 − 2t2 cos ⬘ + 1 共46兲
FIG. 8. 共Color online兲 Periodic slits arranged in a rectangular lattice.
uf共x1,x2兲 =
␣f1 U共x1,x2兲 − f 1x1 t1
¯ ∀ 共x1,x2兲 苸 Y \ ⍀
is a solution of Eq. 共17兲, and therefore,
冤 冥
␣t3 0 1 e A = t1 . k0 0 1
共45兲
The reader is invited to carry out the detailed calculations. More, from Ref. 18 we find
冕冑 冕冑 ⬁
␣ = 
0 ⬁ 0
dt
t + 2t cos 0 + 1 4
2
dt
t4 − 2t2 cos 0 + 1
,
e2 − e1 exp共2i0兲 = , e2 − e3
where 0 ⬍ 0 , ⬘ ⬍ . To investigate the effects of different lattices or unit cells, we compare the following two configurations 共see Figs. 8 and 9兲. It is not hard to see that the periodic slits in Fig. 8 correspond to a rectangular unit cell in Fig. 2 with ␣ = L / 2,  = d / 2, and ␣0 = l / 2, whereas the periodic slits in Fig. 9 correspond to a rhombic unit cell in Fig. 3 with ␣ = L / 4,  = d / 2, and ␣0 = l / 4. Meanwhile, if we shift to the left by L / 2 every other layer of the slits in Fig. 8, we obtain the configuration in Fig. 9. Using Eqs. 共35兲 and 共46兲, in Fig. 4 we plot the effective conductivities along x1-direction of these two configurations against l / L by unmarked curve and “+” markers, respectively. From up to down, the geometric parameters in Figs. 8 and 9 are chosen as d = 2, l = 1, and L = 2.5, 2 , 1.5, 1. We observe that there is no discernible difference between the unmarked curve and + markers in Fig. 4, which means the effective conductivities of the two configurations in Figs. 8 and 9 are the same, at least to the extent of our numerical resolution. The reader may speculate that this arises from symmetry; the author remarks that elementary arguments by symmetry cannot prove this unexpected result. For the case k1 = 0, by similar arguments we verify that uf共x1,x2兲 =
f 2 V共x1,x2兲 − f 2x2 t3
¯ 共47兲 ∀ 共x1,x2兲 苸 Y \ ⍀
satisfies all of Eq. 共19兲. Parallel to Eqs. 共37兲–共40兲 or simply by the duality transformation, we can show the effective con-
FIG. 9. 共Color online兲 Periodic slits arranged in a rhombic lattice.
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103503-9
J. Appl. Phys. 105, 103503 共2009兲
Liping Liu
ductivity is given by Eq. 共40兲 with ␣, , t1, and t3 interpreted as in Eq. 共45兲. V. SUMMARY AND DISCUSSION
We derive a closed-form formula for the effective conductivity of composites with periodic E-inclusion microstructure. When the periodic E-inclusion degenerates to a periodic array of slits, we give explicit solutions to local fields and the effective conductivity of the composite with singular inhomogeneities. Through a linear transformation, these results can be extended to two-phase composites of any anisotropic materials. The results of this paper can be used in the following ways. In the first place the closed-form formula 共13兲 with the volume fraction and the shape matrix Q as parameters can be used to give a quick estimate of the effective properties of a composite. In reality, of course, it is questionable to what extent the microstructure of a composite can be approximated by a periodic E-inclusion. However our prediction 关see Eq. 共13兲兴 is at least physical and realizable, and the qualitative feature of how the effective properties depend on the volume fraction and shape of the inhomogeneities should remain regardless of the exact shapes of the inhomogeneities. Second, the analytic results provide a benchmark for testing various empirical models and numerical codes. Last but not least important, the results in Sec. IV, in particular the comparison between rectangular and rhombic unit cells 共see Fig. 4兲, suggest that the effective properties of composites with a singular phase are predominantly determined by the distance
between nearby inhomogeneities. This has been observed by various authors and is in fact the basis of the network model.6–9 What is noteworthy here is that this remains to be true even for an extreme shape such as a slit, for which the field is not localized between slits 共see Fig. 7兲. D. J. Bergman, Phys. Rep., Phys. Lett. 43, 377 共1978兲. R. Hill, J. Mech. Phys. Solids 13, 213 共1965兲. 3 Z. Hashin and S. Shtrikman, J. Appl. Phys. 33, 3125 共1962兲. 4 J. B. Keller, J. Appl. Phys. 34, 991 共1963兲. 5 A. Dykne, Sov. Phys. JETP 32, 63 共1971兲. 6 V. Ambegaokar, B. I. Halperin, and J. S. Langer, Phys. Rev. B 4, 2612 共1971兲. 7 J. P. Clerc, G. Giraud, J. M. Laugier, and J. M. Luck, Adv. Phys. 39, 191–309 共1990兲. 8 S. M. Kozlov, Russ. Math. Surv. 44, 91 共1989兲. 9 L. Berlyand and A. Kolpakov, Arch. Ration. Mech. Anal. 159, 179 共2001兲. 10 W. R. Delameter, G. Herrmann, and D. M. Barnett, ASME Trans. J. Appl. Mech. 42, 74 共1975兲. 11 J. Wang, J. Fang, and B. L. Karihaloo, Int. J. Solids Struct. 37, 4261 共2000兲. 12 S. B. Vigdergauz, Inzh. Zh., Mekh. Tverd. Tela 21, 165–169 共1986兲. 13 L. P. Liu, R. D. James, and P. H. Leo, Arch. Ration. Mech. Anal. 共unpublished兲. 14 A. Friedman, Variational Principles and Free Boundary Problems 共Wiley, New York, 1982兲. 15 G. W. Milton, The Theory of Composites 共Cambridge University Press, Cambridge, 2002兲. 16 L. P. Liu, R. D. James, and P. H. Leo, Metall. Mater. Trans. A 38, 781 共2007兲. 17 Y. Grabovsky and R. V. Kohn, J. Mech. Phys. Solids 43, 949 共1995兲. 18 A. I. Markushevich, Theory of Function of Complex Variable 共Chelsea, New York, 1977兲, Vol. 1–3. 19 L. V. Ahlfors, Complex Analysis 共McGraw-Hill, New York, 1979兲. 1 2
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