Royal Society Publishing

Expansion formulae for the homogenized determinant of anisotropic checkerboards

Marc Briane, Yves Capdeboscq

Abstract

In this paper, some effective properties of anisotropic four-phase periodic checkerboards are studied in two-dimensional electrostatics. An explicit low-contrast second-order expansion for the determinant of the effective conductivity is given. In the case of a two-phase checkerboard with commuting conductivities, the expansion reduces to an explicit formula for the effective determinant (valid for any contrast) as soon as the second-order term vanishes. Such an explicit formula cannot be extended to four-phase checkerboards. A counter-example with high-contrast conductivities is provided. The construction of the counter-example is based on a factorization principle, due to Astala & Nesi, which allows us to pass from an anisotropic four-phase square checkerboard to an isotropic one with the same effective determinant.

Keywords:

1. Introduction

This paper deals with the effective properties of composites with varying conductivity in two-dimensional electrostatics, in the special case where the conductivity is periodic and where the pattern reproduced periodically is a square. Consider the following conduction problem in a bounded domain Embedded Image of Embedded Image:Embedded ImageThe solution Embedded Image is the voltage potential and f is the density of electric charges. The conductivity Embedded Image is a highly oscillating sequence of the form Embedded Image, where A is a Embedded Image-periodic matrix-valued function. Then, the effective (constant) matrix Embedded Image is the conductivity of the asymptotic problem satisfied by the limit potential as ϵ tends to zero (see formula (2.2) and definition 3.1). We specialize to the case where each square cell is composed of four anisotropic phases:Embedded Image

The conductivity A is constant in each phase of value Embedded Image. Since A may be anisotropic, each Embedded Image is a two-by-two symmetric positive definite matrix. From a mathematical point of view, the effective properties are deduced from the ϵ-rescaled periodic microstructure as the period ϵ tends to zero through a homogenization process (see Bensoussan et al. 1978 for an introduction of the homogenization theory). Our aim is to obtain explicit formulae involving the effective or homogenized matrix. In fact, we restrict ourselves to one coefficient: the determinant of the effective matrix (simply called the effective or homogenized determinant in the sequel).

The effective properties of periodic composites have been widely studied, especially in the case of two-dimensional two-phase composites. After the seminal works of Rayleigh (1892) and Maxwell (1904), Keller (1964) introduced a duality method in order to characterize the effective properties of such composites. His work was extended by Dykhne (1970), Mendelson (1975) and more recently by Golden & Milton (1990). Moreover, explicit solutions were obtained by Berdichevski (1985) and Obnosov (1999) for checkerboard structures, and by Mityushev (1995) for cylindrical inclusions. As an alternative to the derivation of explicit formulae, there is a considerable amount of works on the bounds for, or the approximation of effective coefficients; we refer to Milton (2002 and references therein) for a quite complete review on the bounds theory. On the other hand, numerical results for the effective conductivity of checkerboards were also obtained by Michel et al. (1999) and Torquato et al. (1999). One of the motivations for deriving explicit effective coefficients is for the validation of the numerical approaches.

There are very few explicit formulae for effective coefficients for periodic composites. In particular, there is one result for four-phase structures that explicitly yields the effective coefficients of an isotropic four-phase square checkerboard. In this isotropic setting, each conductivity Embedded Image is equal to Embedded Image, where Embedded Image and Embedded Image is the identity matrix of Embedded Image. The formula was conjectured by Mortola & Steffé (1985), and was proved 15 years later by Craster & Obnosov (2001) for a rectangular checkerboard, and independently by Milton (2001) for a square checkerboard. In the case of a four-phase rectangular checkerboard

Embedded Image

the formula of the effective matrix Embedded Image is rather complicated, but its determinant is given by the following formula:Embedded Image(1.1)

The simplicity of this formula illustrates that, amongst two-dimensional effective constants, the determinant presents particular properties. There are other results that specifically involve the effective determinant. For example, it is known (e.g. Francfort & Murat 1987) that the conductivity matrix of any two-dimensional microstructure (possibly non-periodic) with a constant determinant, induces by homogenization an effective conductivity matrix with the same determinant.

Taking into account the previous results and remarks, we tried to extend the determinant formula (1.1) to an anisotropic four-phase square checkerboard. We did not find a general explicit formula of the effective determinant for such an anisotropic composite. However, assuming that the four phases Embedded Image of the checkerboard admit a second-order expansion around a given matrix Embedded Image, we obtain an explicit second-order expansion for the effective determinant only in terms of Embedded Image and Embedded Image:Embedded Image(1.2)where Embedded Image is the effective matrix of the checkerboard, Embedded Image and Embedded Image are explicit functions of the four phases, and Embedded Image is an explicit function of the reference matrix Embedded Image.

In the case of an anisotropic two-phase checkerboard with commuting conductivity matrices (Embedded Image and Embedded Image with Embedded Image), we prove that the expansion (1.2) gives the exact effective determinant, i.e. Embedded Image, if the second-order term Embedded Image is zero. This leads to the explicit formula Embedded Image.

The situation is much more delicate in the case of an anisotropic four-phase checkerboard. We then restrict ourselves to diagonal conductivity matrices. In §3, we introduce a stability property which connects the effective determinant of the checkerboard with phases Embedded Image to the one with phases Embedded Image, for any positive definite diagonal matrix B. Using the Craster–Obnosov formula (Craster & Obnosov 2001) we check that any isotropic four-phase checkerboard satisfies the stability property. Assuming this property for an anisotropic four-phase checkerboard leads to the expansion (1.2). But in contrast to the case of two-phase checkerboards, the condition Embedded Image does not imply that the correct effective determinant is obtained, i.e. in general Embedded Image.

To prove this negative result, we build an anisotropic four-phase square checkerboard with high-contrast conductivities, which both satisfies Embedded Image and Embedded Image. The counter-example is based on a nice factorization principle due to K. Astala & V. Nesi (2002, personal communication) (see §5). This principle allows us to deduce an anisotropic four-phase square checkerboard from an irregular but isotropic one with the same determinant.

The paper is organized as follows. In §2, we prove an explicit expansion of the effective determinant for an anisotropic two-phase square checkerboard. Section 3 is devoted to a stability property in the general framework of periodic composites. In §4, we study the case of an anisotropic four-phase square checkerboard. Section 5 is devoted to the counter-example.

2. Anisotropic two-phase checkerboards

This section is devoted to anisotropic two-phase square checkerboards. Under a low-contrast assumption between the two phases, the Tartar (1990) small-amplitude homogenization formula allows us to write a second-order expansion of the effective coefficients. This expansion does not provide simple information on the whole homogenized matrix. Indeed, all the coefficients (that is 18 independent coefficients) of the expansions of the two phases appear in the final expansion of the effective matrix.

However, assuming that the conductivity matrices of the two phases commute, the expansion restricted to the effective determinant reduces to an explicit formula in the two phases. Moreover, it is remarkable that the zero-order term of this expansion gives the right effective determinant when the second-order term is zero.

(a) Statement of the result

  1. Embedded Image is the unit square of Embedded Image;

  2. Embedded Image (resp. Embedded Image) is the set of the functions Embedded Image in Embedded Image (resp. Embedded Image) and Y-periodic, i.e. Embedded Image a.e. Embedded Image;

  3. for Embedded Image, Embedded Image is the set of the Y-periodic and symmetric matrix-valued functions A such that, for all Embedded Image,Embedded Image(2.1)

  4. for each Embedded Image, the homogenized matrix associated with the matrix A is denoted by Embedded Image and is defined by the following formula (e.g. Bensoussan et al. 1978), for all Embedded Image,Embedded Image(2.2)

In this section we are interested by an anisotropic two-phase checkerboard structure. Let Embedded Image be two symmetric positive definite matrices of Embedded Image and let A be the Y-periodic matrix-valued function defined byEmbedded Image(2.3)In view of computing the determinant of the homogenized matrix of the two-phase checkerboard (2.3), we have the following result:

  1. Let Embedded Image be a symmetric positive definite matrix of Embedded Image. Let A1, A2 be two symmetric positive definite matrices of Embedded Image such that Embedded Image and which admit the expansion Embedded Image, Embedded Image, around Embedded Image. Then, the homogenized matrix Embedded Image of the checkerboard (2.3) satisfies, if Embedded Image,Embedded Image(2.4)If Embedded Image, Embedded Image. For any positive definite matrix Embedded Image,Embedded Image(2.5)where Embedded Image and where I is the set of all odd integers.

  2. The expansion (2.4) characterizes exactly how the homogenized determinant differs from the simple formula Embedded Image. From this perspective, it is optimal because for any symmetric positive definite matrices Embedded Image such that Embedded Image, we haveEmbedded Image(2.6)

Note that the term in factor of Embedded Image in (2.4) is a second-order term with respect to δ. Part (ii) provides an explicit formula of the effective determinant, which is apparently unrelated to the expansion introduced in part (i). However, the positivity of Embedded Image shows that the converse implication of (2.6) holds true in expansion (2.4) up to higher order terms.

(b) Proof of theorem 2.1

The proof is based on a small-amplitude homogenization formula due to Tartar (1990). More precisely, theorem 1.1, example 2.1 and theorem 4.2 in Tartar (1990) imply the following result.

Let Embedded Image be a Y-periodic matrix-valued function which admits the following second-order expansion around the symmetric positive definite matrix Embedded Image:Embedded Image(2.7)Then, the homogenized matrix Embedded Image defined by (2.2) satisfiesEmbedded Image(2.8)and the correction matrix M is defined byEmbedded Image(2.9)

In the present case, the matrix-valued functions Embedded Image of (2.7) have the two-phase checkerboard structure (2.3) and the corresponding two-phases satisfy the second-order expansion Embedded Image for Embedded Image. The Fourier coefficients of B (with the two-phases Embedded Image) are given byEmbedded ImageDenoteEmbedded ImageThen, putting the value of Embedded Image in formula (2.9) yields the coefficients Embedded Image and Embedded Image of the correction matrix M:Embedded Image(2.10)where Embedded Image are the seriesEmbedded Image(2.11)Taking into account the equalities Embedded Image and Embedded Image, we can compute by means of Maple the second-order expansion of Embedded Image from formulae (2.8) and (2.10) which reads as Embedded Image, where X is given by a very long formula depending on the coefficients of Embedded Image for Embedded Image. Alternatively, we also haveEmbedded ImageFactorizing the terms X and Y with Maple yields Embedded Image if Embedded Image and X=0 otherwise. A lengthy but straightforward computation shows that Embedded Image can be written in a simple form, formula (2.5). This concludes the proof of expansion (2.4). ▪

The proof is based upon a duality argument introduced by Keller (1964). Let Embedded Image be the checkerboard structure obtained by exchanging the phases Embedded Image and Embedded Image. The matrix Embedded Image corresponds to the same structure as that of A up to a translation of vector Embedded Image, and thus yields the same homogenized matrix Embedded Image. We will now find the best constant k>0 such that Embedded Image a.e. in Y. Let Embedded Image and Embedded Image be the eigenvalues of Embedded Image for Embedded Image, with respect to the same basis of eigenvectors. We may then writeEmbedded ImageThe best choice is thus Embedded Image, which in turns implies Embedded Image a.e. in Y. Thanks to a result due to Mendelson 1975 (see also Nevard & Keller 1985; Francfort & Murat 1987), we haveEmbedded Image(2.12)Moreover, definition (2.2) implies that Embedded Image. We thus deduceEmbedded Imagewhich implies Embedded Image, i.e. Embedded Image. Replacing A by Embedded Image also yieldsEmbedded ImageTherefore, we obtainEmbedded Image(2.13)On the other hand, we haveEmbedded ImageIn conclusion we have the following

  1. If Embedded Image then (2.13) implies that Embedded Image.

  2. If Embedded Image then Embedded Image. Whence by a result of Dykhne (1970), Embedded Image and we still obtain Embedded Image.

  3. If Embedded Image then again by Dykhne, Embedded Image. ▪

3. Computation of the homogenized determinant of some microstructures

In this section, we introduce a stability principle in order to compute the effective determinant for some periodic composites. To this end, we study the effects of axial distortions (with respect to the Embedded Image and Embedded Image axes) on the effective properties of a given composite. The stability property then means that the effective determinant of the modified composite (under an axial distortion) reads as the product of the distortion by the effective determinant of the initial composite, without any other interaction.

For example, laminated composites (whose conductivity depends on one direction) and isotropic four-phase rectangular checkerboards (studied in Craster & Obnosov 2001) satisfy the stability property. More generally, for any periodic composite satisfying the stability property we obtain an explicit formula for the effective determinant. In fact, two formulae are derived corresponding to the two axial distortions. This approach by stability will also allow us to construct an explicit second-order expansion for anisotropic four-phase square checkerboards in §4.

(a) A stability under axial deformation property

In the sequel, Embedded Image is a bounded open subset of Embedded Image, Embedded Image, and Embedded Image, for Embedded Image, is the set of the symmetric invertible matrix-valued functions A which satisfy (2.1) on Embedded Image. We shall make use of the theoretical approach to homogenization introduced by Murat & Tartar (1978, 1997), the H-convergence.

A sequence Embedded Image of Embedded Image is said to H-converge to a matrix-valued Embedded Image if for any f in Embedded Image, the solution Embedded Image in Embedded Image of Embedded Image in Embedded Image, satisfies the weak convergencesEmbedded Imagewhere Embedded Image is the solution of Embedded Image in Embedded Image.

The matrix-valued Embedded Image in (3.1) is called the H-limit of Embedded Image and also belongs to the set Embedded Image. We shall always assume that Embedded Image is a sequence of Embedded Image. We shall also use the following notation convention: the H-limit of a positive definite sequence of matrices Embedded Image is noted Embedded Image, and Embedded Image corresponds to the H-limit of the sequence Embedded Image.

Let Embedded Image be a sequence of positive definite matrix-valued functions, which H-converges to Embedded Image. The limit microstructure corresponding to Embedded Image is said to be stable under deformation if for any constant symmetric positive definite matrix B, we haveEmbedded Image(3.1)It is said to be stable under axial deformation if the property holds for any constant diagonal positive definite matrix B.

Note that Embedded Image is defined a priori up to the extraction of a subsequence. In that way, Embedded Image could correspond to subsequence dependent H-limits. In such a case, the definition of stability under deformation is that (3.1) stands for all subsequences.

The choice of the word deformation in this definition is explained by the following proposition.

For any sequence of positive definite matrix-valued functions Embedded Image converging to an H-limit Embedded Image and any constant positive definite matrix B, we have Embedded Image a.e. in Embedded Image, where Embedded Image is the H-limit of the sequence Embedded Image. So, a limit microstructure with constant H-limit is stable under deformation if and only ifEmbedded Image(3.2)

Let us note Embedded Image the H-limit Embedded Image. Consider the Dirichlet problemEmbedded Imagefor some Embedded Image. After an integration by parts against a test function ϕ we obtainEmbedded Image(3.3)which is also, after the change of variable y=Bx, and the notations Embedded Image, Embedded Image,Embedded Image(3.4)Passing to the limit as ϵ tends to 0 in (3.3) we obtainEmbedded Imagewhere Embedded Image is the weak limit of Embedded Image in Embedded Image. Alternatively, passing to the limit in (3.4) we obtainEmbedded ImageEmbedded ImageBy the uniqueness of the limit problem, we have Embedded Image. ▪

This result is known in the general case where Embedded Image with Embedded Image a diffeomorphism (e.g. Tartar 2000). The following proposition gives examples of microstructures which are stable under (axial) deformation.

  1. Any isotropic laminated microstructure is stable under deformation, in any dimension.

  2. More generally, multipliable microstructures (in the sense of Fabre & Mossino 1998) are stable under deformation.

  3. In two dimension, any isotropic four-phase checkerboard is stable under axial deformation.

In the sequel, unless otherwise specified, we will simply write that a matrix or its corresponding microstructure is stable to indicate that it is stable under axial deformation.

(a) For Embedded Image, let Embedded Image be a symmetric matrix, with Embedded Image. It is well known (the original proof being from Murat & Tartar 1978, 1997) that up to the extraction of a subsequence, the homogenized matrix Embedded Image is given byEmbedded Imageand, for all λ such that Embedded Image,Embedded ImageIf Embedded Image is an isotropic matrix, Embedded Image, with Embedded Image, and B be a positive definite symmetric matrix, the above formulae simplifies into Embedded Image, Embedded Image, andEmbedded ImageEmbedded Imagewith Embedded Image, and where Embedded Image (resp. Embedded Image) is the harmonic (resp. arithmetic) average of Embedded Image. We thus obtain Embedded Image, which proves the stability by deformation.

(b) For a multipliable microstructure Embedded Image, that is, for which there exists Embedded Image and Embedded Image such that

  1. Embedded Image with Embedded Image, Embedded Image, Embedded Image, where Embedded Image is denoted by Embedded Image, Embedded Image,

  2. Embedded Image and Embedded Image converge, respectively, to M and P in Embedded Image,

  3. M is invertible.

Fabre & Mossino then showed that Embedded Image. Note that, for a constant positive definite matrix B, the microstructure Embedded Image is also multipliable since the matrices Embedded Image and Embedded Image satisfy the requirements. Clearly, Embedded Image converges to Embedded Image and Embedded Image converges to Embedded Image. Invoking again the Fabre–Mossino result, Embedded Image, and in particular, A is stable under deformation.

(c) Let us now turn to the case of an isotropic two-dimensional four-phase periodic checkerboard, that is, with A defined byEmbedded Imageand repeated periodically. Note that, since it is a periodic structure, the H-limit is constant, and by (3.2) the invariance by axial deformation amounts to Embedded Image. For any positive definite diagonal matrix Embedded Image, the microstructure corresponding to Embedded Image corresponds to a four-phase (of equal area) rectangular Embedded Image checkerboard. In both cases, the homogenized matrices Embedded Image and Embedded Image have been obtained by Craster & Obnosov (2001, p. 8), and their common determinant isEmbedded Image ▪

(b) Computation of the homogenized determinant for a stable two-dimensional microstructure

In order to state this result, we introduce the following notation: for an integrable and Y-periodic function f of one or two variables, we note Embedded Image the arithmetic average of f with respect to the ith variable:Embedded Image

Let Embedded Image be a stable Embedded Image-periodic microstructure. Then, the corresponding effective determinant is given by the two following formulae:Embedded Image(3.5)

Embedded Image(3.6)

If A is diagonal, formulae (3.5) and (3.6) simplify intoEmbedded Image

By a rotation of angle Embedded Image of the periodic pattern, formula (3.5) yields formula (3.6). We will thus prove (3.5). Let Embedded Image be a Y-periodic microstructure. For Embedded Image, let Embedded Image be defined as Embedded Image. Note that Embedded Image, where B is the diagonal positive definite matrix with entries p and q. By hypothesis, A is stable and therefore Embedded Image for all positive p, q.

Passing to the limit in p and q, we will obtain (3.5). We first assume that q is fixed. Since A and in turn Embedded Image are Y-periodic, the homogenized matrix Embedded Image is given in terms of its correctors Embedded Image byEmbedded Image(3.7)with the notation Embedded Image and where Embedded Image is the unique solution in Embedded Image of the cell problemEmbedded Image(3.8)Note that the sequence Embedded Image is bounded in Embedded Image uniformly in p, and therefore up to a subsequence, converges weakly to a limit Embedded Image. Hence (see Allaire 1992), there exists a function Embedded Image in Embedded Image such that, up to a subsequence, Embedded Image two-scale converges to Embedded Image. Note that the matrices Embedded Image converges strongly in the sense of two-scale convergence towards Embedded Image, that isEmbedded ImageSo, for any Embedded Image and Embedded Image, Embedded Image is an admissible test function, that converges strongly in the sense of two-scale convergence to its two-scale limit. Multiplying (3.8) by Embedded Image and integrating by parts, we obtainEmbedded ImagePassing to the limit as p tends to Embedded Image, we obtainEmbedded Image(3.9)and by (3.7)Embedded Image(3.10)By density, equation (3.9) holds true for all Embedded Image. Computing the corrector Embedded Image which is the classic corrector for laminates in the Embedded Image direction, we obtain that Embedded Image is solution of Embedded Image in Embedded Image and that (3.10) simplifies intoEmbedded Image(3.11)where Embedded Image is the function of the second variable given by Embedded Image, Embedded Image, Embedded Image and Embedded Image. Note thatEmbedded Image(3.12)We now consider Embedded Image as an oscillating sequence of period Embedded Image. The same arguments applies and we can again pass to the limit in q in formula (3.11). A new homogenized matrix appears Embedded Image which is constant. As a consequence,Embedded ImageThe determinant of Embedded Image can be obtained from (3.12), exchanging the roles of the first and second indexes, and substituting Embedded Image for A. This givesEmbedded Imagewhich is (3.5). ▪

For a diagonal periodic matrix A, taking the test function Embedded Image depending on the first (or second) variable only in the homogenized formula (2.2) yieldsEmbedded ImageUsing (as in the proof of theorem 2.1) the identity (2.12) we also obtainEmbedded ImageFormulae (3.6) are the products of these upper and lower bounds. In the special case when the microstructure A is diagonal with separable variables, i.e. Embedded Image, Embedded Image, the four inequalities above are equalities (in such a case, the structure is multipliable, as it is explained in proposition 3.4).

4. Checkerboard with four anisotropic phases

In §3, we noted that any isotropic four-phase checkerboard is stable in the sense of (3.2). It is then natural to ask if anisotropic checkerboards are stable. We restrict ourselves to a checkerboard with four diagonal phases Embedded Image, Embedded Image (see figure 1), whose periodic matrix is defined in the unit square Y byEmbedded Image(4.1)where Embedded Image, Embedded Image, Embedded Image and Embedded Image. We still denote by Embedded Image its homogenized matrix. Omitting to verify that the microstructure A is stable under axial deformation, we can compute the values of the homogenized determinants Embedded Image and Embedded Image given by (3.6). We obtainEmbedded Image(4.2)Both formulae are equal if Embedded Image with Embedded Image, Embedded Image andEmbedded Image(4.3)It is natural to wonder whether formula (4.2) provides an approximation of the determinant in the general case. We have the following asymptotic result.

Figure 1

A four-phase checkerboard.

Let Embedded Image be a positive diagonal matrix of Embedded Image. Let Embedded Image, Embedded Image, be four positive diagonal matrices of Embedded Image, which admit around Embedded Image the expansion Embedded Image, Embedded Image. Then, the homogenized matrix Embedded Image of the four-phase checkerboard (4.1) A is given byEmbedded Image(4.4)where Embedded Image is the determinant given by (4.2), E is given by (4.3) and Embedded Image reads asEmbedded Imagewhere Embedded Image and where the sum is taken over all odd numbers.

The difference Embedded Image corresponds to the distance between the two determinants Embedded Image and Embedded Image obtained as limits by deformation. It is remarkable that, once corrected of this difference scaled by the constant factor Embedded Image, the candidate homogenized determinant Embedded Image is valid up to the second order.

There are several examples for which the asymptotic stability condition Embedded Image gives the correct determinant:

  1. As already mentioned, in the case of an isotropic checkerboard, Embedded Image and Embedded Image.

  2. In the case of laminates of diagonal matrices, that is, whenEmbedded Imagewhich implies Embedded Image, the microstructure A is multipliable (see proposition 3.4) and thus Embedded Image.

  3. In the case of a two-phase anisotropic checkerboard structure, the asymptotic stability condition is optimal. Indeed, if Embedded Image then an easy computation yieldsEmbedded Imagewhence Embedded Image by theorem 2.1.

The following theorem (which will be proved in §5a) shows that it is not the case in general for four-phase anisotropic checkerboards:

There exists a four-phase anisotropic checkerboard with Embedded Image and Embedded Image.

Therefore, contrary to what was obtained for two-phase checkerboards, the second-order term Embedded Image of expansion (4.4) cannot be used as an indicator of the simplicity or complexity of the formula of the effective determinant. In that sense and in contrast to expansion (2.4), expansion (4.4) is not optimal.

The proof is similar to that of theorem 2.1. By the definition of the four-phase checkerboard matrix-valued A defined by (4.1), the Fourier coefficients of the first-order term B in (2.7) with its four phases Embedded Image, Embedded Image, are given byEmbedded Imagewhere I is the set of the odd integers. The correction matrix M defined by (2.9) thus satisfiesEmbedded Image(4.5)where S, T are the seriesEmbedded Image(4.6)Then, we expand up to second order the determinant of Embedded Image using Tartar's formula (2.8) with the correction matrix (4.5), and the expression Embedded Image defined by (4.2). After simplifications of formulae by means of Maple we compare the second-order terms of each expansion and we obtain the desired result (4.4). ▪

5. A high-contrast counter-example

The following counter-example is based on a factorization principle introduced by Astala & Nesi (personal communication) in order to treat anisotropic periodic composites. The principle consists in making a change of variable in which the new conductivity is still periodic (with a new period) but isotropic. The gradient of the change of variable corresponds to the electric field associated with the rescaled conductivity obtained by dividing the initial one by the square root of its determinant; so, the rescaled conductivity is still anisotropic but its determinant is equal to 1. The new effective conductivity (obtained from the new isotropic composite) is deduced from the old one (obtained from the initial anisotropic composite); thanks to a factorization formula such that the new and old effective determinants are equal. Therefore, the change of variable allows them to pass from an anisotropic periodic composite to an isotropic one without change of the effective determinant.

In the case of a four-phase checkerboard with diagonal conductivity matrices, we construct a change of variable which reduces to a piecewise linear function, linear in each of the regions where the phase is constant. Combining the constraints satisfied by the electric field, the potential and the current at the interfaces between phases (respectively, periodicity, continuity and continuity of the normal derivative) we obtain a linear system in the coefficients of the piecewise linear function. This system has a non-trivial solution if the eight conductivity coefficients (for the four diagonal phases) satisfy a particular condition. When this conditions holds, the change of variable leads to a new periodic composite, with constant coefficients on an irregular but isotropic four-phase checkerboard. We then propose a suitable choice of high-contrast conductivities such that the new effective conductivity satisfies all the constraints, but such that the explicit formula of §4 does not hold for the effective determinant.

(a) A result from Astala & Nesi

This section is devoted to the proof of theorem 4.3. We will construct a family of microstructures Embedded Image, Embedded Image, satisfying the asymptotic stability condition Embedded Image, with E is given by (4.3) and such Embedded Image for τ close enough to 1. We will rely on the following result of Alessandrini & Nesi (2001, 2002).

Let A be a Y-periodic matrix-valued function in Embedded Image. Define Embedded Image a.e. in Y. Let Embedded Image be a function in Embedded Image such that Embedded Image is Y-periodic and which solves the equation Embedded Image in Embedded Image. Assume that the Y-averaged value of Embedded Image is a non-zero vector. Then, there exists a stream function ψ in Embedded Image such thatEmbedded Imagewhich solves the conjugate equation Embedded Image in Embedded Image. Moreover, the matrix-valued function Embedded Image is an homeomorphism from Embedded Image onto Embedded Image and the Y-averaged value of Embedded Image is an invertible matrix.

We define Embedded Image, where Embedded Image. By the conditions satisfied by Embedded Image and ψ, the new variable Embedded Image also reads asEmbedded Image(5.1)where B is a constant invertible matrix and Embedded Image a Y-periodic vector-valued function in Embedded Image. In the new coordinate system, Embedded Image is periodic of period BY and is isotropic. Then, the following factorization result is due to Astala & Nesi (personal communication):

Using the above notations, if Embedded Image is the H-limit of Embedded Image and Embedded Image the H-limit of Embedded Image, thenEmbedded Image(5.2)

For the reader convenience we just give an idea of the proof following Milton's approach as described in §§8.5 and 8.6 of Milton (2002).

On the one hand, using the definition of Embedded Image in terms of Embedded Image and ψ yieldsEmbedded ImageOn the other hand, the curvilinear form (5.1) of the change of variable Embedded Image leads to Embedded Image, which in particular implies (5.2). ▪

(b) Application to the four-phase anisotropic checkerboard

In order to use the result of Astala & Nesi (personal communication), we construct a piecewise linear change of variable Embedded Image satisfying the conditions given in theorem 5.1, i.e. for Embedded Image, Embedded Image, Embedded Image and Embedded Image. Functions Embedded Image and ψ are continuous and periodic, which yields 16 equations at the interfaces between the quadrants of Y, repeated periodically. The flux condition Embedded Image provides another eight identities. It is convenient to writeEmbedded ImageIf Embedded Image, these 24 equations imply Embedded Image for Embedded Image, violating the assumption of theorem 5.1, i.e. Embedded Image. We obtain on Embedded Image and Embedded ImageEmbedded ImageThe columns of both matrices are orthogonal, thus Embedded Image is an orthogonal change of variable. Furthermore, the continuity along the interface yields Embedded Image and Embedded Image. Therefore, Embedded Image and Embedded Image are two rectangles with a common side. The quadrant Embedded Image has lengthEmbedded Imagewhereas Embedded Image has lengthEmbedded Image

Performing the same analysis on the other adjacent quadrants, we obtain that Y maps onto Embedded Image the rectangle represented in figure 2. Since Embedded Image is an arbitrary parameter, we set Embedded Image. The corresponding matrixEmbedded Imageis represented in figure 2. We can check that the Y-averaged valued of Embedded Image is also a non-zero vector. Therefore, according to theorem 5.1, Embedded Image and Embedded Image have the same determinant.

Figure 2

The new period cell Embedded Image, and the microstructure Embedded Image. Here, Embedded Image and Embedded Image for Embedded Image.

In the case when all four phases have equal area, i.e. Embedded Image, the matrix-valued function Embedded Image is an isotropic four-phase checkerboard; its homogenized matrix is given in Craster & Obnosov (2001). This happens when Embedded Image, that is, when Embedded Image for all Embedded Image and for some positive α. In this case, the simpler (linear) change of variable Embedded Image, with Embedded Image, leads directly to the isotropic checkerboard; thanks to proposition 3.3. Using the Craster–Obnosov result we also obtainEmbedded Image

(c) Proof of theorem 4.3

Consider the four-phase checkerboard (4.1) given byEmbedded ImageUsing the same notations as above,Embedded ImageWe do have Embedded Image. Therefore, Embedded Image, whereEmbedded ImageThe domain Embedded Image is a square of side Embedded Image, and the quadrant Embedded Image is also a square of side Embedded Image. Since Embedded Image, we have Embedded Image. In this case, formula (4.4) reads asEmbedded Image(5.3)We will prove that Embedded Image cannot always be equal to Embedded Image. First, note that we can transform Embedded Image to a square of side 1, by a homothetic transformation (which does not change the value of the homogenized matrix). The new geometry is represented.Embedded Image

On the one hand, by letting Embedded Image tend to zero we obtain a structure where sub-square Embedded Image is not conducting. Note that the homogenized matrix of the structure obtained when Embedded Image is set to zero is same as the one obtained passing to the limit when Embedded Image tends to zero. When the volume fraction is small, the formula for Embedded Image is that of a ‘Virtual Mass’. Jikov et al. (1994, pp. 106–107) proved thatEmbedded Imagewhere Embedded Image is a symmetric positive definite matrix.

On the other hand, when Embedded Image tends to zero the conjectured determinant Embedded Image defined by (5.3) satisfiesEmbedded ImageTherefore, Embedded Image for τ close to 1.

6. Conclusion

This contribution points out that the effective properties of an anisotropic four-phase periodic checkerboard can be partially but explicitly attained through the expansion of the effective determinant. Indeed, for a two-phase checkerboard, a second-order expansion formula provides an explicit approximation of the effective determinant, which is valid if the conductivity matrices commute and have a weak contrast. A remarkable fact is that the exact value for the effective determinant is obtained when the second-order term of the expansion vanishes. In that sense, this expansion is somewhat optimal.

In the more general case of a four-phase checkerboard, the stability property introduced in §3 holds true for any isotropic four-phase rectangular checkerboard; thanks to the Craster–Obnosov formula. This property allows us to derive an explicit second-order expansion for a four-phase checkerboard with diagonal conductivity matrices. Unfortunately, in contrast to the two-phase checkerboard case, the cancellation of the second-order term of the expansion does not imply that the exact value of the effective determinant is given by the expansion, as it is shown by the counter-example of §5.

The counter-example is interesting in itself since it provides an application of a nice factorization principle for anisotropic periodic composites, due to Astala & Nesi (personal communication). In that example, an explicit construction of piecewise linear functions in each phase allows us to transform an anisotropic four-phase square checkerboard into an irregular but isotropic one with the same effective determinant.

Amongst the initial motivations for this work was the desire to evaluate an automatic formula generation algorithm. Both positive and negative results presented in this article can be exploited to this end. Moreover, it could be used to design (analytical) benchmarks for anisotropic homogenization numerical codes. Alternatively, the approach based on the stability property is not restricted to four-phase checkerboards. It could be exploited to obtain explicit expansion formulae of the effective determinant for other periodic composites.

Acknowledgements

The authors wish to thank V. Nesi for a stimulating discussion and relevant comments concerning the derivation of the counter-example. The authors were partly supported by ACINIM 2003-45.

Footnotes

    • Received January 13, 2006.
    • Accepted February 9, 2006.

References

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