## 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.

## 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 of :The solution is the voltage potential and *f* is the density of electric charges. The conductivity is a highly oscillating sequence of the form , where *A* is a -periodic matrix-valued function. Then, the effective (constant) matrix 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:

The conductivity *A* is constant in each phase of value . Since *A* may be anisotropic, each 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 is equal to , where and is the identity matrix of . 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

the formula of the effective matrix is rather complicated, but its determinant is given by the following formula:(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 of the checkerboard admit a second-order expansion around a given matrix , we obtain an explicit second-order expansion for the effective determinant only in terms of and :(1.2)where is the effective matrix of the checkerboard, and are explicit functions of the four phases, and is an explicit function of the reference matrix .

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

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 to the one with phases , 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 does not imply that the correct effective determinant is obtained, i.e. in general .

To prove this negative result, we build an anisotropic four-phase square checkerboard with high-contrast conductivities, which both satisfies and . 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

is the unit square of ;

(resp. ) is the set of the functions in (resp. ) and

*Y*-periodic, i.e. a.e. ;for , is the set of the

*Y*-periodic and symmetric matrix-valued functions*A*such that, for all ,(2.1)for each , the homogenized matrix associated with the matrix

*A*is denoted by and is defined by the following formula (e.g. Bensoussan*et al*. 1978), for all ,(2.2)

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

*Let**be a symmetric positive definite matrix of*.*Let A*_{1}, A_{2}be two symmetric positive definite matrices of*such that**and which admit the expansion*,*, around*.*Then, the homogenized matrix**of the checkerboard**(2.3)**satisfies, if**,*(2.4)*If**,*.*For any positive definite matrix**,*(2.5)*where**and where I is the set of all odd integers*.*The expansion**(2.4)**characterizes exactly how the homogenized determinant differs from the simple formula*.*From this perspective, it is optimal because for any symmetric positive definite matrices**such that**, we have*(2.6)

Note that the term in factor of 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 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* *be a Y-periodic matrix-valued function which admits the following second-order expansion around the symmetric positive definite matrix* *:*(2.7)*Then, the homogenized matrix* *defined by* *(2.2)* *satisfies*(2.8)*and the correction matrix M is defined by*(2.9)

In the present case, the matrix-valued functions of (2.7) have the two-phase checkerboard structure (2.3) and the corresponding two-phases satisfy the second-order expansion for . The Fourier coefficients of *B* (with the two-phases ) are given byDenoteThen, putting the value of in formula (2.9) yields the coefficients and of the correction matrix *M*:(2.10)where are the series(2.11)Taking into account the equalities and , we can compute by means of Maple the second-order expansion of from formulae (2.8) and (2.10) which reads as , where *X* is given by a very long formula depending on the coefficients of for . Alternatively, we also haveFactorizing the terms *X* and *Y* with Maple yields if and *X*=0 otherwise. A lengthy but straightforward computation shows that 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 be the checkerboard structure obtained by exchanging the phases and . The matrix corresponds to the same structure as that of *A* up to a translation of vector , and thus yields the same homogenized matrix . We will now find the best constant *k*>0 such that a.e. in *Y*. Let and be the eigenvalues of for , with respect to the same basis of eigenvectors. We may then writeThe best choice is thus , which in turns implies a.e. in *Y*. Thanks to a result due to Mendelson 1975 (see also Nevard & Keller 1985; Francfort & Murat 1987), we have(2.12)Moreover, definition (2.2) implies that . We thus deducewhich implies , i.e. . Replacing *A* by also yieldsTherefore, we obtain(2.13)On the other hand, we haveIn conclusion we have the following

If then (2.13) implies that .

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

If then again by Dykhne, . ▪

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

A sequence of is said to *H*-converge to a matrix-valued if for any *f* in , the solution in of in , satisfies the weak convergenceswhere is the solution of in .

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

Let be a sequence of positive definite matrix-valued functions, which *H*-converges to . The limit microstructure corresponding to is said to be stable under deformation if for any constant symmetric positive definite matrix *B*, we have(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 is defined *a priori* up to the extraction of a subsequence. In that way, 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* *converging to an H-limit* *and any constant positive definite matrix B, we have* *a.e. in* *, where* *is the H-limit of the sequence* . *So, a limit microstructure with constant H-limit is stable under deformation if and only if*(3.2)

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

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

*Any isotropic laminated microstructure is stable under deformation, in any dimension*.*More generally, multipliable microstructures (in the sense of**Fabre & Mossino 1998**) are stable under deformation*.*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 , let be a symmetric matrix, with . It is well known (the original proof being from Murat & Tartar 1978, 1997) that up to the extraction of a subsequence, the homogenized matrix is given byand, for all *λ* such that ,If is an isotropic matrix, , with , and *B* be a positive definite symmetric matrix, the above formulae simplifies into , , andwith , and where (resp. ) is the harmonic (resp. arithmetic) average of . We thus obtain , which proves the stability by deformation.

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

with , , , where is denoted by , ,

and converge, respectively, to

*M*and*P*in ,*M*is invertible.

Fabre & Mossino then showed that . Note that, for a constant positive definite matrix *B*, the microstructure is also multipliable since the matrices and satisfy the requirements. Clearly, converges to and converges to . Invoking again the Fabre–Mossino result, , 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 byand 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 . For any positive definite diagonal matrix , the microstructure corresponding to corresponds to a four-phase (of equal area) rectangular checkerboard. In both cases, the homogenized matrices and have been obtained by Craster & Obnosov (2001, p. 8), and their common determinant is ▪

### (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 the arithmetic average of *f* with respect to the *i*th variable:

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

(3.6)

If *A* is diagonal, formulae (3.5) and (3.6) simplify into

By a rotation of angle of the periodic pattern, formula (3.5) yields formula (3.6). We will thus prove (3.5). Let be a *Y*-periodic microstructure. For , let be defined as . Note that , where *B* is the diagonal positive definite matrix with entries *p* and *q*. By hypothesis, *A* is stable and therefore 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 are *Y*-periodic, the homogenized matrix is given in terms of its correctors by(3.7)with the notation and where is the unique solution in of the cell problem(3.8)Note that the sequence is bounded in uniformly in *p*, and therefore up to a subsequence, converges weakly to a limit . Hence (see Allaire 1992), there exists a function in such that, up to a subsequence, two-scale converges to . Note that the matrices converges strongly in the sense of two-scale convergence towards , that isSo, for any and , is an admissible test function, that converges strongly in the sense of two-scale convergence to its two-scale limit. Multiplying (3.8) by and integrating by parts, we obtainPassing to the limit as *p* tends to , we obtain(3.9)and by (3.7)(3.10)By density, equation (3.9) holds true for all . Computing the corrector which is the classic corrector for laminates in the direction, we obtain that is solution of in and that (3.10) simplifies into(3.11)where is the function of the second variable given by , , and . Note that(3.12)We now consider as an oscillating sequence of period . The same arguments applies and we can again pass to the limit in *q* in formula (3.11). A new homogenized matrix appears which is constant. As a consequence,The determinant of can be obtained from (3.12), exchanging the roles of the first and second indexes, and substituting for *A*. This giveswhich is (3.5). ▪

For a diagonal periodic matrix *A*, taking the test function depending on the first (or second) variable only in the homogenized formula (2.2) yieldsUsing (as in the proof of theorem 2.1) the identity (2.12) we also obtainFormulae (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. , , 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 , (see figure 1), whose periodic matrix is defined in the unit square *Y* by(4.1)where , , and . We still denote by its homogenized matrix. Omitting to verify that the microstructure *A* is stable under axial deformation, we can compute the values of the homogenized determinants and given by (3.6). We obtain(4.2)Both formulae are equal if with , and(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.

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

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

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

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

In the case of laminates of diagonal matrices, that is, whenwhich implies , the microstructure

*A*is multipliable (see proposition 3.4) and thus .In the case of a two-phase anisotropic checkerboard structure, the asymptotic stability condition is optimal. Indeed, if then an easy computation yieldswhence by theorem 2.1.

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

*There exists a four-phase anisotropic checkerboard with* *and* .

Therefore, contrary to what was obtained for two-phase checkerboards, the second-order term 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 , , are given bywhere *I* is the set of the odd integers. The correction matrix *M* defined by (2.9) thus satisfies(4.5)where *S*, *T* are the series(4.6)Then, we expand up to second order the determinant of using Tartar's formula (2.8) with the correction matrix (4.5), and the expression 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 , , satisfying the asymptotic stability condition , with *E* is given by (4.3) and such 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* . *Define* *a.e. in Y*. *Let* *be a function in* *such that* *is Y-periodic and which solves the equation* *in* . *Assume that the Y-averaged value of* *is a non-zero vector*. *Then, there exists a stream function ψ in* *such that**which solves the conjugate equation* *in* . *Moreover, the matrix-valued function* *is an homeomorphism from* *onto* *and the Y-averaged value of* *is an invertible matrix*.

We define , where . By the conditions satisfied by and *ψ*, the new variable also reads as(5.1)where *B* is a constant invertible matrix and a *Y*-periodic vector-valued function in . In the new coordinate system, 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* *is the H-limit of* *and* *the H-limit of* *, then*(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 in terms of and *ψ* yieldsOn the other hand, the curvilinear form (5.1) of the change of variable leads to , 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 satisfying the conditions given in theorem 5.1, i.e. for , , and . Functions and *ψ* are continuous and periodic, which yields 16 equations at the interfaces between the quadrants of *Y*, repeated periodically. The flux condition provides another eight identities. It is convenient to writeIf , these 24 equations imply for , violating the assumption of theorem 5.1, i.e. . We obtain on and The columns of both matrices are orthogonal, thus is an orthogonal change of variable. Furthermore, the continuity along the interface yields and . Therefore, and are two rectangles with a common side. The quadrant has lengthwhereas has length

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

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

### (c) Proof of theorem 4.3

Consider the four-phase checkerboard (4.1) given byUsing the same notations as above,We do have . Therefore, , whereThe domain is a square of side , and the quadrant is also a square of side . Since , we have . In this case, formula (4.4) reads as(5.3)We will prove that cannot always be equal to . First, note that we can transform 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.

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

On the other hand, when tends to zero the conjectured determinant defined by (5.3) satisfiesTherefore, 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.

- © 2006 The Royal Society