effective properties of some new self-similar structures

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Effective properties of some new self-similar structuresDag Lukkassen a & Annette Meidell aa Narvik University College , Narvik, NorwayPublished online: 21 Sep 2010.

To cite this article: Dag Lukkassen & Annette Meidell (2008) Effective properties of some newself-similar structures, Applicable Analysis: An International Journal, 87:12, 1297-1310, DOI:10.1080/00036810802140624

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Applicable AnalysisVol. 87, No. 12, December 2008, 1297–1310

Effective properties of some new self-similar structures

Dag Lukkassen* and Annette Meidell

Narvik University College, Narvik, Norway

Communicated by A. Pankov

(Received 22 December 2007; final version received 30 March 2008)

We consider a self-similar structure of chessboard type with infinitely manymicro-scales where the conductivity is locally anisotropic. The effective propertiesare found by using �-convergence techniques. Even though the effectiveconductivity matrix coincide with that of the standard chessboard structure inthe isotropic case, we show that this is generally not the case. Our results are usedto construct sequences of two-component material-structures, whose effectiveconductivities are described by the well-known arithmetic-geometric meanintroduced by Legendre and Gauss.

Keywords: effective properties; reiterated homogenization; self-similar structures

1. Introduction

In this article, we consider a two-component, locally anisotropic material defined onself-similar structures with infinitely many micro-levels. We find a simple formula for thecorresponding effective conductivity matrix by studying the limit behaviour of energyfunctionals as the fineness of the microstructure increases at all micro-levels. This is doneby means of �-convergence, a well-known concept which was introduced by De Giorgi andSpagnolo in the late 60s for the mathematical study of the macroscopic properties ofcomposite materials. We also utilize the concept of reiterated homogenization, which wasintroduced on the physical level by Bruggeman already in the 30s and justifiedmathematically by Bensoussan, Lions and Papanicolaou in 1978.

When the material is locally isotropic it turns out that our formula coincide with theDykhne–Keller formula for standard chessboards. One might think that these twostructures have equal effective properties in general. However, even if the duality trick forobtaining the Dykhne–Keller formula is useful also for certain locally anisotropic cases,it cannot be used to obtain simple algebraic expression for the effective conductivity matrixof standard chessboards in all cases. Moreover, by establishing bounds we are able to findthe limit value of the effective conductivity of a standard chessboard in the horizontaldirection when the local conductivity in the vertical direction approaches 0. This value

*Corresponding author. Email: [email protected]

ISSN 0003–6811 print/ISSN 1563–504X online

� 2008 Taylor & Francis

DOI: 10.1080/00036810802140624

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turns out to be different from the one obtained from our formula, and we can therefore

rule out the possibility that these two structures have equal effective properties in general.Our results on the self-similar structure may be used to find new structures with

interesting effective properties. As an example, we construct sequences of two-component

material-structures whose effective conductivities converge to the well-known arithmetic-

geometric mean and the harmonic-geometric mean (of Legendre and Gauss) in the vertical

and horizontal direction, respectively.

2. Preliminary results on !-convergence and reiterated homogenization

Let us consider the class A of Lagrangians g :Rn�R

n!R of the form g(x, �)¼ � �B(x)�,

where B(�) is a measurable matrix and

c1 �j j2� gðx, �Þ � c2 �j j

2, ð1Þ

where 05 c1� c251. If gh and g1 belong to this class, g1 is said to be the �-limit of the

sequence gh, denoted by g1¼�� lim gh, if for any open bounded set ��Rn with

Lipschitz boundary the following two conditions hold:

(i) for any uh2W1,2(�), uh* u weakly in W1,2(�) it holds thatZ

�

g1ðx,DuÞdx � lim infh!1

Z�

ghðx,DuhÞdx,

(ii) for every u2W1,2(�) there is a sequence uh such that uh* u weakly inW1,2(�) and

uh � u 2W1, 20 ð�Þ, such thatZ

�

g1ðx,DuÞdx ¼ limh!1

Z�

ghðx,DuhÞdx:

This type of convergence is particularly useful in the study of macroscopic properties

of composite materials where the integer h measures the fineness of the microstructure

(the larger h, the finer is the microstructure). For more information we refer to the

literature (see, e.g. the books [1] and [2]). An important result is that the class A is compact

with respect to �-convergence.As an example, consider the case when ��R

2 forms the cross-section of a locally

isotropic composite bar with local shear modulus �h¼�h(x). The corresponding torsional

rigidity Dh is then given by

Dh ¼ minv2H1ð�Þ

FhðvÞ,

where

FhðvÞ ¼

Z�

grad vþ ð�x2, x1Þð Þ � Bh grad vþ ð�x2, x1Þð Þdx

¼

Z�

�h@v

@x� x2

� �2

þ�h@v

@yþ x1

� �2

dx,

(Bh¼�hI ). If the sequence gh(x, �)¼ � �Bh(x)� �-converges to some Lagrangian

g(x, �)¼ � �B1(x)�, it turns out that Dh!D. Here, D is the torsional rigidity of an

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anisotropic rod of the same cross-section � with out-of-plane shear modulus matrix of the

form

B1 ¼C2323 C2313

C1323 C1313

" #,

where {Cijkl} denotes the corresponding fourth-order elasticity tensor. Hence,

D ¼ minv2H1ð�Þ

FðvÞ,

where

FðvÞ ¼

Z�

grad vþ ð�x2, x1Þð Þ � B1 grad vþ ð�x2, x1Þð Þdx:

Also, note that if Bh(x) is the local heat conductivity matrix and gh(x, �)¼ � �Bh(x)��-converges to some Lagrangian g(x, �)¼ � �B1(x)�, we say that Bh(x) G-converges to

B1(x), denoted by BhðxÞ*GB1ðxÞ, and call B1(x) the corresponding effective conductivity

matrix.The most classical examples of �-converging Lagrangians gh(x, �)¼ � �Bh(x)� are the

ones of the form

BhðxÞ ¼ Bðhx, . . . , hMxÞ, ð2Þ

where B is Y-periodic and piecewise continuous in each of the M variables and satisfies

the condition (1). In this case it turns out that the �� lim gh exists, and we denote this

limit by gB.

THEOREM 1 If Bh(x) is of the form (2) then the �-limit gB¼�� lim gh exists and is

independent of x. Moreover, gB¼ g[M] is found iteratively according to the following scheme:

g m�k½ �ðx1, . . . , xk, �Þ ¼ minu2W1, 2

per ðYÞ

ZY

g m�k�1½ �ðx1, . . . ,xk, y, � þDuð yÞÞdy,

g 0½ �ðx1, . . . , xM, �Þ ¼ � � Bðx1, . . . , xMÞ�:

In this article, we will consider cases when B is of the form

Bðx1, . . . , xMÞ ¼ Dðx1Þ þ �A1ðx1ÞðDðx2Þ þ �A1

ðx2Þ Dðx3Þ þ �A1ðx3Þ Dðx4Þ þ � � �ð

�þ �A1

ðxMÞD1

��: ð3Þ

Here, Dð yÞ ¼PM

i¼2 Dk�Akð yÞ, D1, . . . ,DM are constant symmetric positive definite

matrixes and A1, . . . ,Am are (sufficiently smooth) Y-periodic regions such that the

Lebesgue measure of the set R2n (A1[ � � � [Am) is zero (as usual �A denotes the

characteristic function of the set A).More generally we may consider Lagrangians ghðx, �Þ ¼ � � bBhðxÞ� of the form

bBhðxÞ ¼ Bhðhx, . . . , hMxÞ, ð4Þ

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where Bh is of the form

Bhðx1, . . . , xMÞ ¼ Dhðx1Þ þ �A1ðx1ÞðDhðx2Þ

þ �A1ðx2Þ Dhðx3Þ þ �A1

ðx3Þ Dhðx4Þ þ � � � þ �A1ðxMÞD1, hðxMÞ

� �� : ð5Þ

Here, Dhð yÞ ¼PM

k¼2 Dk, hð yÞ�Akð yÞ, where Dk, h( y) are symmetric matrixes satisfying

the property

�1 �j j2� � �Dk, hð yÞ� � �2 �j j

2,

for some constants �1, �2, 05 �1��251 with the additional property that Dk, h*GDk

as h!1.

THEOREM 2 Let B be defined by (3). It holds that

�� lim gh ¼ gB,

where gB is given in Theorem 1.

The first variant of Theorem 1 was proved by Bensoussan, Lions and Papanicolaou in

the book [3]. Various generalizations can be found in [1,4–8]. However, for non-standard

Lagrangians satisfying the growth condition

�c0 þ c1 �j jq� gðx1, . . . , xm, �Þ � c0 þ c2 �j j

p, ð6Þ

the �-limit may fail to exist if m4 1. In this case, the best we can achieve is obtaining

optimal upper and lower bounds for the limit-Lagrangians of all �-converging

subsequences. These bounding Lagrangians can be found iteratively by a more general

scheme than the above one (see [9]). Concerning the historical development of reiterated

homogenization and its application we refer to the survey-articles [10] and [11]. The proof

of Theorem 2 can be found in [12].We also want to mention that optimal design problems of composites (mixture of

chessboard structures, cellular materials and laminates) were considered in the papers of

Lurie and Cherkaev [13].

3. A self-similar structure of chessboard type

Put Y¼ [�1, 1]2 and let �?, �b and �w be the indicator functions defined on Y by

�?ðxÞ ¼1 if x 2 �1, 0½ �

2[ 0, 1½ �

2,

0 otherwise ,

(

�bðxÞ ¼1 if x 2 �1, 0½ � � 0, 1½ �,

0 otherwise ,

�

�wðxÞ ¼1 if x 2 0, 1½ � � �1, 0½ �,

0 otherwise ,

�


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and extended Y-periodically to R2. Given any positive integer h, we may construct a two-

component media with conductivity matrixes

Db ¼�b, 1 0

0 �b, 2

" #and Dw ¼

�w, 1 0

0 �w, 2

" #as follows. Let

Dð yÞ ¼�1ð yÞ 0

0 �2ð yÞ

" #, ð7Þ

where �i( y)¼�b( y)�b,iþ�w( y)�w,i, and let the conductivity matrix at the point x2R2

be given by

BhðxÞ ¼ DðhxÞ þ �?ðhxÞðDðh2xÞ þ �?ðh

2xÞ Dðh3xÞ þ �?ðh3xÞ Dðh4xÞ þ � � ��

Þ

¼ DðhxÞ þX1m¼1

�mk¼1�?ðh

kxÞ� �

Dðhmþ1xÞ:ð8Þ

The underlying microstructure has infinitely many self-similar micro-levels of chessboard

type (Figure 1).

THEOREM 3 If Bh(x) is given by (8) then BhðxÞ*GB1ðxÞ, where B1 is the constant matrix

given by

B1 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�b, 1�w, 1p

0

0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�b, 2�w, 2p

" #: ð9Þ

Proof Let Y¼ [�1, 1]2 and let ’f be the �-limit of the sequence fh(x, �)¼ f(hx, �), wheref ðx, �Þ ¼

P2i¼1 �iðxÞj�ij

2 and �i, i¼ 1, 2 are measurable and Y-periodic functions such that

05�� i��5þ1 for some constants � and �. Note that this limit exists according to

Theorem 1. It is possible to show that

’�f ð�Þ � ’fð�Þ � ’þf ð�Þ, ð10Þ

Figure 1. A self-similar structure with infinitely many micro-levels of chessboard type.


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where

’�f ð�Þ ¼X2i¼1

q�i �ið Þ�2i , ’þf ð�Þ ¼

X2i¼1

qþi �ið Þ�2i , ð11Þ

and q�i ð�iÞ, qþi ð�iÞ are the harmonic-arithmetic means defined by

q�i �ið Þ ¼1

2

Z 1

�1

1

2

Z 1

�1

�ið Þ�1 dxi

� ��1dxj, ð12Þ

qþi �ið Þ ¼1

2

Z 1

�1

1

2

Z 1

�1

�i dxj

� ��1dxi

!�1, ð13Þ

where i, j2 {1, 2} and i 6¼ j (Remark 4).Let �min¼min {�b,1,�w,1,�b,2,�w,2}, �max¼max {�b,1,�w,1,�b,2,�w,2} and let the

Lagrangians �i ðx, �Þ and f�i ðx, �Þ, i2 {1, 2} be constructed according to the following

iterative formulae:

�i ðx, �Þ ¼X2j¼1

�jðxÞ�2j þ �?ðxÞ’ �

i�1ð�Þ, ’ �

0ð�Þ ¼ �min

X2j¼1

�2j , ð14Þ

þi ðx, �Þ ¼X2j¼1

�jðxÞ�2j þ �?ðxÞ’ þ

i�1ð�Þ, ’ þ

0ð�Þ ¼ �max

X2j¼1

�2j , ð15Þ

f �i ðx, �Þ ¼X2j¼1

�jðxÞ�2j þ �?ðxÞ’

�f �i�1ð�Þ, ’ �f �

0ð�Þ ¼ �min

X2j¼1

�2j , ð16Þ

f þi ðx, �Þ ¼X2i¼j

�jðxÞ�2j þ �?ðxÞ’

þ

f þi�1

ð�Þ, ’ þf þ0

ð�Þ ¼ �max

X2j¼1

�2j : ð17Þ

Iterated use of (10) and the fact that �min��max give the inequalities

’ þf �Mð�Þ � ’ �

Mð�Þ � ’ þ

Mð�Þ � ’ þ

f þM

ð�Þ: ð18Þ

Moreover, by applying the general formulae (11–13) to the Lagrangians f �i ðx, �Þ we find

that ’ �f �ið�Þ ¼

P2j¼1 �

�j, i�

2j and ’ þ

f þi

ð�Þ ¼P2

j¼1 �þj, i�

2j , where ��j, i are the numbers found

iteratively as follows:

��j, i ¼

��1b, j þ ��j, i�1

� �1� �2

� �� 1þ ��1w, j þ ��j, i�1

� �1� �2

� �� 12

, � �j, 0 ¼ �min,


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�þj, i ¼

�b, j þ �þj, i�1

� 2

� �� 1þ �w, j þ �

þj, i�1

� 2

� �� 12

0BBB@1CCCA�1

, � þj, 0 ¼ �max: ð19Þ

It is fairly easy to see that �min � ��j, i�1 � �

�j, i � �max and �min � �

þj, i � �

þj, i�1 � �max, i.e.

the sequences f��j, ig and f�þj, ig are monotone and bounded, and therefore convergent with

limits ��j and �þj , respectively, satisfying the equations

��j ¼

��1b, j þ ��j

� �1� �2

� �� 1þ ��1w, j þ ��j

� �1� �2

� �� 12

,

�þj ¼

�b, j þ �þj

� 2

� �� 1þ �w, j þ �

þj

� 2

� �� 12

0BBB@1CCCA�1

:

These equations turn out to have the same solutions, namely � �j ¼ �þj ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�b, j�w, jp

(this is

our key-observation!). Hence, we deduce from (18) that

limM!1

’ �Mð�Þ ¼ lim

M!1’ þ

Mð�Þ ¼ s1ð�Þ ¼

X2j¼1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�b, j�w, jp

�2j : ð20Þ

Motivated by the fact that

shðx, �Þ ¼X2j¼1

�2j ½�jðhxÞ þ �?ðhxÞð�jðh2xÞ þ �?ðh

2xÞð. . . ,

we introduce the Lagrangians s �M, hðx, �Þ and s þM, hðx, �Þ (with M scales) defined by

s �M, hðx, �Þ ¼X2j¼1

�2j �jðhxÞ þ �?ðhxÞð�jðh

2xÞ þ �?ðh2xÞ

� � � � þ �?ðhM�1xÞ �jðh

MxÞ þ �?ðhMxÞ�min

� ��

� ��,

s þM, hðx, �Þ ¼X2j¼1

�2j �jðhxÞ þ �?ðhxÞð�jðh

2xÞ þ �?ðh2xÞ

� � � � þ �?ðhM�1xÞ �jðh

MxÞ þ �?ðhMxÞ�max

� ��

� ��:

and let S �Mð�Þ and S þMð�Þ denote the corresponding �-limits. Theorem 1 gives that

S �Mð�Þ ¼ ’ �M ð�Þ, S þMð�Þ ¼ ’ þM ð�Þ: ð21Þ


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By compactness, there exists a subsequence sh 0 of sh such that �� lim sh 0 ¼ S for some

Lagrangian S. Since

s �M, hðx, �Þ � shðx, �Þ � s þM, hðx, �Þ,

it is clear that

’ �Mð�Þ ¼ S �Mð�Þ � Sðx, �Þ � S þMð�Þ ¼ ’ þM ð�Þ:

Hence, by (20) we obtain that S(x, �)¼ s1(�).This shows that every �-converging subsequence of sh converges to s1, which implies

that the whole sequence {sh} �-converges to s1. Indeed, let u2W1,2(�), let {vh} be such

that vh* u weakly in W1,2(�) and

limh 0!1

Z�

sh 0 ðx,Dvh 0 Þdx ¼ infuh*u

lim infh!1

Z�

shðx,DuhÞdx,

for some subsequence {h0} of {h}. Due to compactness and the above result there exists

a subsequence fsh 00 g of fsh 0 g such that �þ lim sh 00 ¼ s1. Thus, we easily see thatZ�

s1ðDuÞdx ¼ infuh*u

lim infh!1

Z�

shðx,DuhÞdx:

In a similar way we can prove thatZ�

s1ðDuÞdx ¼ infuh*u

lim suph!1

Z�

shðx,DuhÞdx:

Thus,

infuh*u

lim infh!1

Z�

shðx,DuhÞdx ¼ infuh*u

lim suph!1

Z�

shðx,DuhÞdx:

This shows that �� lim sh¼ s1 for the whole sequence.

Remark 4 The proof of the bounds (12) and (13) are almost identical with that given in

([14]). These bounds can also be deduced from some more general bounds obtained in [15],

whose proof is more complicated.

4. A more generalized structure

For application purposes we will need a slightly more general structure obtained by

changing the matrix D( y) given in (7) with

Dhð yÞ ¼�1, hð yÞ 0

0 �2, hð yÞ

� �, �i,hð yÞ ¼ �bð yÞ�b, i, hð yÞ þ �wð yÞ�w, i, hð yÞ, ð22Þ

where ��b, i, h�� and ��w,i,h�� are such that

Dh!G �b, j 0

0 �w, j

" #:


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THEOREM 5 Let Bh(x) be given as in (8) with the only difference that the matrix D( y) is

replaced with Dh( y) given by (22). Then, the sequence sh(x, �)¼ � �Bh(x)� �-converges to

s1ð�Þ ¼ � � B1�, where B1 is the constant matrix given by

B1 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�b, 1�

w, 1

p0

0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�b, 2�

w, 2

p" #: ð23Þ

Proof The proof is very similar to that of Theorem 3. Indeed, by replacing �b, j, �w, j with

�b, j, �w, j before Equtaion (20), replacing �j( y) with �j, h( y) in the rest of that proof,

and using Theorem 2 instead of Theorem 1 to obtain (21), all arguments remain valid,

and the result follows directly.

5. The standard chessboard structure

The standard chessboard structure ( Figure 2) is much easier to describe than the above

structure. Let � be the indicator function defined by

�ðxÞ ¼1 if x1x2 4 0,

0 if x1x2 5 0,

�

Figure 2. The standard chessboard structure.


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on Y¼ [�1, 1]2 and extended Y-periodically to R2. Moreover, let the conductivity matrix

at the point x2R2 be given by Bh(x)¼B(hx), where

Bð yÞ ¼�1ð yÞ 0

0 �2ð yÞ

� �:

and �i( y)¼�( y)�b, iþ (1��( y))�w, i. In the special case of local isotropy, i.e. when

�b,1¼�b,2¼�b and �w,1¼�w,2¼�w, it was shown by Keller [16] and Dykhne [17] that the

effective conductivity matrix B1 ¼ffiffiffiffiffiffiffiffiffiffiffi�b�wp

I, where I denotes the identity matrix,

i.e. it coincides with that of the self-similar structure of Section 3. The general anisotropic

case appears still to be open. However, in view of the similarity between these structures

for the isotropic case one might expect that B1 generally takes the form (9) also

for standard chessboard structures. However, the following result shows that this is not

the case.

THEOREM 6 Let B1 be the effective conductivity matrix for the standard chessboard

structures. Then, if �b,2! 0 and �w,2! 0 it holds that

B1 !

2�b, 1�w, 1

�b, 1 þ �w, 10

0 0

24 35:Proof Since �i( y) is symmetric in the x1 and x2 direction, we know that B1 is diagonal

(compare, e.g. with [2, p. 39]), i.e. of the form

B1 ¼�1 0

0 �2

� �:

By the Reuss–Voigt–Wiener bounds (see, e.g. [2]) it holds that

X2i¼1

�ið�Þð Þ�1

� ��1�2i �

X2i¼1

�i �2i �

X2i¼1

�ið�Þð Þ� �

�2i ,

where h�i denotes the average h�i ¼ jYj�1RY (�)dx, i.e.

2�b, i�w, i

�b, i þ �w, i� �i �

�b, i þ �w, i

2: ð24Þ

This shows that �2 ! 0 as �b,2! 0 and �w,2! 0. By Theorem 1 we find that

�1 ¼ minu2W1, 2

per ðYÞf ðuÞ,

f ðuÞ ¼1

Yj j

ZY

�1ðxÞ 1þ@u

@x1

� �2

þ�2ðxÞ@u

@x2

� �2 !

dx:


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Let �4 0 and let w 2W1, 2perðYÞ be of the form

wðx1,x2Þ ¼

sðx1Þ � � x2 � 1,

rðx1, x2Þ 0 � x2 � �,

hðx1Þ �1þ � � x2 � 0,

tðx1, x2Þ �1 � x2 � �1þ �,

8>>><>>>:where r(x1, x2) and t(x1, x2) are linear interpolations between s(x1) and h(x1),

e.g. (r(x1, x2)� h(x1))/(x2þ �)¼ (s(x1)� h(x1))/�. In the region

rðx1, x2Þ � hðx1Þ ¼x2�þ 1

� sðx1Þ � hðx1Þð Þ,

Yrt ¼ x 2 Y : �1 � x2 � �1þ � or 0 � x2 � �f g

we find that

�2ðxÞ @w=@x2ð Þ2¼ �2ðxÞ sðx1Þ � hðx1Þð Þ

2=�2:

Hence, choosing �24�2(x) and letting �! 0, we obtain thatZYrt

�2ðxÞ@w

@x2

� �2

dx! 0:

We also find that ZYrt

�1ðxÞ@w

@x1þ 1

� �2

dx! 0,

as �! 0. Thus,

f ðwÞ !1

Yj j

ZY �ð�1ðxÞ 1þ

@hðx1Þ

@x1

� �2 !

dxþ

ZY þð�1ðxÞ 1þ

@sðx1Þ

@x1

� �2!dx

!,

where Y�¼ [�1, 1]� [�1, 0] and Yþ¼ [�1, 1]� [0, 1]. Since �1(x) is only dependent of x1in each of the regions Y� and Yþ separately, the minimum of these values are found

exactly as in the laminate case:ZY �ð�1ðxÞ 1þ

@hðx1Þ

@x1

� �2!dx ¼

ZY þð�1ðxÞ 1þ

@sðx1Þ

@x1

� �2!dx ¼ Y þ

�� 1ð�Þð Þ�1

� ��1:

Hence, in the limit we obtain that

�1 ¼ minu2W1, 2

per ðYÞf ðuÞ � �1ð�Þð Þ

�1� ��1

¼2�b, 1�w, 1

�b, 1 þ �w, 1:

Combined with (24) this gives that �1 ¼ 2�b, 1�w, 1=ð�b, 1 þ �w, 1Þ, and the proof is

complete.


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6. Sequences of structures related to the arithmetic-geometric mean

Based on the self-similar structure above, we can construct a sequence of two-component

material-structures in the following way. We start with two isotropic materials with

conductivities �b and �w (illustrated in Figure 3 with black and white colours,

respectively). Using equal proportions of these two materials we construct a laminate

material-structure, denoted by A1, and a material-structure of the exact same type as that

described in Section 3, denoted by B1. These two material structures are then imbedded

into two new structures with same geometry as the previous ones, and we denote the

resulting structures A2 and B2. These structures are illustrated in Figure 3, where the

colour indicates the type of imbedded structure. In Figure 3, it is certainly understood that

homothetic contractions of the structures A1 and B1 are imbedded into all levels of the ‘?’ -

region of B2. Continuing this process recursively we obtain structures Ai, Bi. The existence

of the corresponding effective conductivity matrices Ai , Bi follows by Theorem 5.

Moreover, by using (9) and the well-known formula for the effective conductivity matrix

corresponding to laminates (see, e.g. [2]), we easily obtain that

Ai ¼ai,1 0

0 ai,2

� �, Bi ¼

bi,1 0

0 bi,2

� �,

where ai, j and bi, j are found by the following iterative formulae:

ai,2 ¼1

2ai�1,2 þ bi�1,2� �

, bi,2 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiai�1,2bi�1,2

p, a0, 2 ¼ �b, b0, 2 ¼ �w, ð25Þ

1

ai,1¼

1

2

1

ai�1,1þ

1

bi�1,1

� �,1

bi,1¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

ai�1,1

1

bi�1,1

s,

1

a0, 1¼

1

�b,

1

b0, 1¼

1

�w. ð26Þ

Figure 3. Sequences of structures {Ai} and {Bi}.


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Since

0 � ai,2 � bi,2 ¼1

2ai�1,2 þ bi�1,2� �

�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiai�1,2bi�1,2

p�

1

2ai�1,2 � bi�1,2� �

,

for i4 1, i.e.

0 � ai,2 � bi,2 �1

2ai�1,2 � bi�1,2� �

,

we see that ai,2 and bi,2 converge to the same limit. This limit is called the arithmetic-

geometric mean agm(�b,�w). The name stems from the fact that we start with two given

numbers �b and �w and repeatedly form arithmetic and geometric mean. Observing that

(25) and (26) are of the same form, we also obtain that a�1i,1 and b�1i,1 converge to

agmð��1b ,��1w Þ: Hence, we conclude that

limi!1

Ai ¼ limi!1

Bi ¼hgmð�b,�wÞ 0

0 agmð�b,�wÞ

� �,

where hgm(�b,�w) is the harmonic-geometric mean,

hgmð�b,�wÞ ¼ agmð��1b ,��1w Þ� ��1

:

The value ðagmð1,ffiffiffi2pÞÞ�1 is known as the Gauss’s constant, and has the closed form

1

agmð1,ffiffiffi2pÞ¼

2

Z 1

0

dtffiffiffiffiffiffiffiffiffiffiffiffi1� t4p ,

which was first observed by Gauss. The arithmetic-geometric mean has many interesting

properties. For example, one can show that

agmða, bÞ ¼

4

aþ b

K ½ða� bÞ=ðaþ bÞ�ð Þ,

where K(x) is the complete elliptic integral of the first kind. Moreover,

agmða, bÞ ¼

Z 1�1

dxffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2 þ a2ð Þ x2 þ b2ð Þ

p !�1:

In addition, it holds that (agm(1þ x, 1� x))�1 and (agm(1, x))�1 are solutions to the

equation

x3 � x� � d2y

dx2þ ð3x2 � 1Þ

dy

dxþ xy ¼ 0:

The convergences of the sequences ai,2 and bi,2 are extremely rapid. The above algorithm

for the construction of these sequences is therefore often used for rapid computation of

K(x) and other related functions.


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References

[1] A. Braides and A. Defranceschi, Homogenization of Multiple Integrals, Oxford University Press,Oxford, 1998.

[2] V.V. Jikov, S.M. Kozlov, and O.A. Oleinik, Homogenization of Differential Operators andIntegral Functionals, Springer-Verlag, Berlin, 1994.

[3] A. Bensoussan, J.L. Lions, and G.C. Papanicolaou, Asymptotic Analysis for Periodic Structures,

Amsterdam, North Holland, 1978.[4] G. Allaire and M. Briane, Multiscale convergence and reiterated homogenization, P. Roy. Soc.

Edinb. 126A (1996), pp. 297–342.[5] A. Braides and D. Lukkassen, Reiterated homogenization of integral functionals, Math. Mod.

Meth. Appl. Sci. 10 (2000), pp. 47–71.[6] D. Lukkassen, Formulae and bounds connected to optimal design and homogenization of partial

differential operators and integral functionals, Ph.D. thesis, Depatment of Mathematics, Tromso

University, Norway, 1996.[7] J.-L. Lions et al., Reiterated homogenization of monotone operators, C. R. Acad. Sci. Paris, Ser. I,

Math. 330 (2000), pp. 675–680.

[8] J.-L. Lions et al., Reiterated homogenization of nonlinear monotone operators, Chin. Ann. Math.,Ser. B 22 (2001), pp. 1–14.

[9] D. Lukkassen, Reiterated homogenization of non-standard Lagrangians, C. R. Acad. Sci., Paris,

Ser. I, Math 332 (2001), pp. 999–1004.[10] D. Lukkassen and G.W. Milton, On hierarchical structures and reiterated homogenization,

Proceedings of the Conference on Function Spaces, Interpolation Theory and Related Topics inHonour of Jaak Peetre on his 65th Birthday, Walter de Gruyter, Berlin, 2002, pp. 311–324,

August 17 – 22, 2000.[11] D. Lukkassen, G. Nguetseng, and P. Wall, Two scale convergence, Int. J. Pure Appl. Math. 2

(2002), pp. 35–86.

[12] D. Lukkassen, A. Meidell, and P. Wall, Multiscale homogenization of monotone operators,Discrete Cont. Dyn.-A. 22 (2008), pp. 711–727.

[13] K.A. Lurie and A.V. Cherkaev, Effective characteriztics of composite materials and the optimal

design of structural elements, Topics in the Mathematical Modelling of Composite Materials,175-258, Progress in Nonlinear Differential Equations and their Applications, Vol. 31,Birkhauser Boston, Boston, MA, 1997.

[14] D. Lukkassen, Some sharp estimates connected to the homogenized p-Laplacian equation,ZAMM-Z.Angew. Math. Mech. 76 (1996), pp. 603–604.

[15] D. Lukkassen, A. Meidell, and P. Wall, Bounds on the effective behaviour of a homogenizedReynold-type equation, J. Funct. Spaces Appl. 5 (2007), pp. 133–150.

[16] J.B. Keller, A theorem on the conductivity of a composite medium, J. Math. Phys 5(4) (1964),pp. 548–549.

[17] A.M. Dychne, Conductivity of a two-phase two-dimensional system, J. Exper. Theor. Phys 59(7)

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effective properties of some new self-similar structures

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