New exact results for the eective electric, elastic, piezoelectric and other properties of composite ellipsoid assemblages

Transcription

1 Journal of the Mechanics and Physics of Solids 51 (2003) New exact results for the eective electric, elastic, piezoelectric and other properties of composite ellipsoid assemblages Y. Benveniste a;, G.W. Milton b a Department of Solid Mechanics, Materials and Systems, Faculty of Engineering, Tel-Aviv University, Ramat-Aviv, Tel Aviv 69978, Israel b Department of Mathematics, University of Utah, E Rm 233, Salt Lake City, Utah 84112, USA Abstract In this paper we present a unied treatment of composite ellipsoid assemblages in the setting of uncoupled phenomena like conductivity and elasticity and coupled phenomena like thermoelectricity and piezomagnetoelectricity. The building block of this microgeometry is a confocal ellipsoidal particle consisting of a (possibly void) core and a coating. All space is lled up with such units which have dierent sizes but possess the same aspect ratios. The confocal ellipsoids may have the same orientation in space or may be randomly oriented. The resulting microgeometry simulates two-phase composites in which the reinforcing components are short bers or elongated particles. Our main interest is in obtaining information of an exact nature on the eective moduli of this microgeometry whose eective tensor symmetry structure depends on the packing mode of the coated ellipsoids. This information will sometimes be complete like the full eective thermoelectric tensor of an assemblage which contains aligned ellipsoids in which the coating is isotropic and the core is arbitrarily anisotropic. In the majority of the cases however the maximum achievable exact information will be only partial and will appear in the form of certain exact relations between the eective moduli of the microgeometry. These exact relations are obtained from exact solutions for the elds in the microstructure for a certain set of loading conditions. In all the considered cases an isotropic coating can be combined with a fully arbitrary core. This covers the most important physical case of anisotropic bers in an isotropic matrix. Allowing anisotropy in the coating requires the fulllment of certain constraint conditions between its moduli. Even though in this case the presence of such constraint conditions may render the anisotropic coating material hypothetical, the value of the derived solutions remains since they still provide benchmark comparisons for approximate and numerical treatments. The remarkable feature of the general analysis which covers all treated uncoupled and coupled Corresponding author. Tel.: ; fax: addresses: benben@eng.tau.ac.il (Y. Benveniste), milton@math.utah.edu (G.W. Milton) /03/$ - see front matter? 2003 Elsevier Ltd. All rights reserved. doi: /s (03)

2 1774 Y. Benveniste, G.W. Milton / J. Mech. Phys. Solids 51 (2003) phenomena is that it is developed solely on the basis of potential solutions of the conduction problem in the same microgeometry.? 2003 Elsevier Ltd. All rights reserved. Keywords: A. Microstructures; A. Voids and inclusions; B. Coupled-eld exact solutions; B. Fibre-reinforced composite material; B. Particulate reinforced material 1. Introduction The present paper is concerned with the study of confocal ellipsoid assemblages in the context of uncoupled and coupled eld phenomena. This microgeometry consists of confocal ellipsoids made up of a core and coating which ll up the whole space and is a generalization of the composite sphere assemblage of Hashin (1962) and Hashin and Shtrikman (1962). In these works, exact expressions for the eective bulk modulus and eective conductivity of the composite sphere assemblage were derived for any volume fraction of the core which remains the same for all coated spheres. It turns out that these expressions provide a good approximation to the corresponding measured eective moduli of actual two-phase composites which consist of spherical particles in an isotropic matrix. In the case of conductivity the derived formula coincides with the well-known approximation of Clausius and Mossotti, thus giving credibility to that approximation. The eective conductivity and bulk modulus of the composite sphere assemblage provide also a realization of the renowned Hashin-Shtrikman bounds and allow thus to prove their optimality. In the case of brous composites, Hashin and Rosen (1964) derived exact expressions for four of the ve moduli of a composite cylinder assemblage which consists of coated circular cylinders and behaves in an eectively transversely isotropic manner. Generalizations of the composite sphere assemblage exist in the literature. Since it turns out that the eld inside the core region of each coated sphere is uniform (as induced by a hydrostatic loading in the case of elasticity and always in the conduction case) one can treat it as an eective medium in which other families of coated spheres could be embedded. Such multi-coated spheres and cylinders have been proposed by Schulgasser (1977), Milton (1981a, b), Lurie and Cherkaev(1985), and Milgrom and Shtrikman (1989). On the other hand, the exploration of confocal ellipsoid assemblages which is the subject of this paper is of course a natural one. Such a microstructure simulates two-phase composites in which the reinforcing components are short bers. As pointed out by Grabovsky and Kohn (1995), it may be also relevant to the equilibrium shapes of coherent precipitates in crystalline solids (see the references cited there in that respect). A comprehensive survey on the existing results on composite ellipsoid assemblages is given in Chapter 7 of the recent book by Milton (2002) (to be denoted by TC throughout the present paper). In the context of conductivity such assemblages have been analyzed by Milton (1981a), Bergman (1982) and Tartar (1985), see also Benveniste and Miloh (1989, 1991) for applications to cracked bodies and coated short ber composites. In the more involved context of elasticity substantial progress has been made by Grabovsky and Kohn (1995) who showed that a suitable

3 Y. Benveniste, G.W. Milton / J. Mech. Phys. Solids 51 (2003) confocal ellipse construction minimizes the elastic energy of a mixture of two isotropic materials with specied volume fractions which is subjected to a given uniform strain in two-dimensions. Most relevant to the present study is the important observation made in that work and proved by Grabovsky (1996) which states that the condition for attainment of this energy bound implies that a solution for the conductivity problem can be used to generate a solution for the elasticity problem in microstructures such as coated ellipsoid assemblages that will achieve the bound (more general conditions for the attainability of the so-called translation bounds can be found in Section 25.5 in TC, referred to as unpublished work in the papers of Grabovsky and Kohn, and Grabovsky). This is explicitly illustrated in Section 7.9 of TC where a study is made of a confocal ellipsoid assemblage in which both the core and coating are elastic and isotropic. It is shown in that study that for a certain uniform strain loading on the microgeometry which is xed by the geometry and the material parameters of the confocal ellipsoid (and which results in a given state of uniform and hydrostatic strain in the core), an exact solution can be found in the coating and thus throughout the assemblage. For a given packing geometry which determines the material symmetry of the eective composite, this allows the establishment of a number of exact relations between the eective moduli of that microgeometry. The remarkable feature of the analysis in that section is that it is based solely on potential solutions of the conduction problem with the same geometry (as expected from the prior results of Grabovsky and Kohn, 1995). Finally it is appropriate to mention here a series of papers in which appear solutions of a confocal ellipsoidal (or elliptical) particle in an innite medium which is subjected at innity to a uniform loading; this conguration is of interest in studying systems of coated particles embedded in a matrix or in formulating the so-called generalized self-consistent method for estimating the eective moduli of a matrix containing non-coated ellipsoidal (or elliptical) inclusions (Mikata and Taya, 1985, 1986; Hatta and Taya, 1986; Miloh and Benveniste, 1988; Huang and Hu, 1995; Ru, 1999; Ru et al., 1999; Riccardi and Montheillet, 1999). The present paper is concerned with the study of composite ellipsoid assemblages in which both the core and coating are anisotropic. Uncoupled phenomena like conductivity and elasticity, as well coupled phenomena like thermoelectricity and piezoelectricity are studied. It should be noted that, composite ellipsoid assemblages with an anisotropic behaviour both in the core and coating have been treated until now in the literature only in the context of a stretched confocal ellipsoid assemblage which is obtained by a suitable linear transformation of the regular composite ellipsoid assemblage with an isotropic coating, see Section 8.4 in TC, and Grabovsky (1996). Moreover, these studies were carried out in the setting of uncoupled phenomena. The establishment of exact results for confocal ellipsoid assemblage microgeometries are expected to possess several merits: (a) They provide a realizable approximation for actual composite media. These microgeometries go a step beyond the simpler lamination constructions, periodic checkerboard representations and phase invariant microgeometries in two-dimensional problems for which exact solutions can be found (see Chapter 9 and Sections 3.2 and 4.7 in TC). (b) Under certain conditions they allow one to prove the optimality of existing bounds for certain two-phase actual composites. (c) They provide benchmark comparative solutions for various approximation schemes.

4 1776 Y. Benveniste, G.W. Milton / J. Mech. Phys. Solids 51 (2003) (d) The exact solution for the neutral inhomogeneity problem which is a building block for the composite ellipsoid assemblages provides a reference check for numerical treatments of inhomogeneity problems in bodies of nite or innite extent. In Section 2, we use a general notation for describing the constitutive behaviour of solids in the context of general coupled elds. All known uncoupled eld phenomena like conductivity, elasticity and coupled eld phenomena like thermoelectricity and piezoelectricity are obtained as a special case of this representation. In Section 3, which is the main section of the paper, an exact solution is constructed for a neutral confocal ellipsoid inhomogeneity in an external host medium. Specically, a coated ellipsoidal inhomogeneity is embedded in the host solid which is subjected to a certain uniform eld on its boundary S (say, a constant electric eld and a constant strain eld in the case of piezoelectricity). We ask whether there exists a possible relation between the properties of the core, coating, external medium, the geometrical parameters of the coated ellipsoid and the nature of the applied external loading so that the eld in the host medium remains undisturbed after the introduction of the inhomogeneity in it. Once established, such an inhomogeneity is called neutral with respect to the applied loading. We show that a positive answer to this question can be found by generalizing the analysis in Section 7.9 in TC which treats the case of an isotropic core and coating in the context of elasticity. Once the solution for the neutral inhomogeneity problem has been achieved, the ground has been set for adding more inhomogeneities until the whole space is lled by them. A proof of the feasibility of lling up the whole space by inhomogeneities of any shape is given in the Appendix to the paper. This proof is an expanded version of the argument which exists in Milton and Serkov(2001). After the composite ellipsoid assemblage has been constructed our analysis provides certain exact relations to be satised by the eective moduli of the assemblage microgeometry. The complete eective moduli tensor for the assembly can be determined only in those cases in which a solution for the neutral inhomogeneity could be constructed for an arbitrary external uniform loading. Sections 4 to 9 consist in the application of the results of Section 3 to several physical phenomena. An important characteristic of the derived solution in Section 3 is that the elds (e.g., electric eld, strains) in the core are uniform. Generally it turns out that a solution for the described neutral inhomogeneity problem becomes possible only if the material parameters of the coating obey certain constraint conditions, and if the external loading is of a certain type. For the case in which the coating is isotropic no constraint condition is present for its material parameters. As the nature of these phenomena become more complicated (i.e., from conductivity to elasticity and to piezoelectricity), and the anisotropy of the coating increases, one gets more constraints on the material parameters of the coating and on the nature of the external loading for which a solution becomes possible. Even though the presence of such constraint conditions in the case of an anisotropic coating may render the material hypothetical, the value of the derived solutions remains as far as providing benchmark comparisons to approximate schemes and numerical treatments. The case of rotational material symmetry of the coating about one of the axis of the ellipsoid is of particular interest. In this case only one constraint condition is obtained in the context of elasticity as well as in n-coupled eld phenomena with scalar potentials (like magnetoelectricity and thermoelectricity). In the

5 Y. Benveniste, G.W. Milton / J. Mech. Phys. Solids 51 (2003) case of piezomagnetoelectricity there are two constraint conditions if there is rotational material symmetry, and one constraint condition if additional reectional symmetries are present. It is of interest to note that in the case of one constraint condition, if the material parameters depend on the temperature, then it may be possible to determine the specic temperature for which that constraint condition on the coating material is satised. Section 4 is concerned with conductivity type problems with the presence of the Hall eect (which allows one to consider conductivity tensors which are not necessarily symmetric), Section 5 with elasticity, Section 6 with n-coupled eld problems with scalar potentials and Section 7 with piezomagnetoelectricity (which encompasses piezoelectricity, piezomagnetism and magnetoelectricity as special cases). In Section 8 we give a general treatment of n-coupled elds in assemblages of confocal elliptical cylinders. In all these sections the coated ellipsoids or elliptical cylinders which make up the assemblage have their axes aligned. Finally, in Section 9 we study randomly oriented composite ellipsoid assemblages with an isotropic coating and anisotropic core which result in an eectively isotropic behaviour. New results are obtained in that section for the eective conductivity and eective bulk modulus of such assemblages. 2. Constitutive behaviour In this paper we use a general notation for describing the constitutive behaviour of linear solids as presented in Chapter 6 in TC. The notation has classically been used to describe coupled elds like magnetoelectricity and thermoelectricity but can be adapted to represent more complex phenomena like piezoelectricity and piezomagnetoelectricity. Dene n divergence free vector elds J 1 (x); J 2 (x);:::;j n (x) and n curl free vector elds E 1 (x); E 2 (x);:::;e n (x) which are linked through the constitutive relation: J i (x)=l ij (x)e j (x); (2.1) where and are eld indices assuming the values 1 to n, whereas i and j are space indices assuming the values 1,2,3. A matrix representation of (2.1) is: J 1 L 11 L 12 : : L 1n E 1 J 2 L 21 L 22 : : L 2n E 2 : : : : : : : = ; (2.2) : : : : : : : : : : : : : : J n L n1 L n2 : : L nn E n where the subscripts refer here to eld indices and the ijth element of the matrix L is L ij. In the case of magnetoelectricity, for example, n = 2 and E 1 ; E 2 denote the electric and magnetic eld, respectively, whereas J 1 ; J 2 denote the electric displacement and magnetic ux.

6 1778 Y. Benveniste, G.W. Milton / J. Mech. Phys. Solids 51 (2003) Let us describe the use of the above notation to represent piezomagnetoelectricity. This coupled eld phenomenon is dened by (see, Alshits et al., 1992, for example) ij = C ijkl kl e kij E k q kij H k ; D i = e ikl kl + ij E j + ij H j ; B i = q ikl kl + ji E j + ij H j ; (2.3) where ij ; ij are the stress and strain tensors, and D i ;E i ;B i ;H i are the electric displacement, electric eld, magnetic ux and magnetic eld vectors, respectively. The elastic properties are denoted by the fourth order stiness tensor C ijkl, whereas the piezoelectric and piezomagnetic properties are denoted by the third order tensors e kij ;q kij. The second order tensors ij ; ij ; ij are the dielectric properties, magnetic permeabilities, and magnetoelectric coecients. The strain, electric eld, and magnetic elds are derivable from a displacement eld u i, electric potential, and magnetic potential : ij = )/ j 2; E i ; H i : i The following symmetry properties prevail for the above dened tensors: ij = ji ; ij = ji ; C ijkl = C jikl = C ijlk = C klij ; e kij = e kji ; q kij = q kji ; ij = ji ; ij = ji : (2.5) In order to cast (2.3) in the form of (2.1), we make use of (2.4) and (2.5) soasto get: i = C ij u ;j + e ji ;j + q ji ;j ; D i = e ij u ;j ij ;j ij ;j ; ;=1; 2; 3 i; j =1; 2; 3 B i = q ij u ;j ji ;j ij ;j ; (2.6) where, at this stage, ; refer to space indices as well. Eq. (2.6) shows that the piezomagnetoelectric law can be written as a coupled eld phenomenon with ve coupled elds dened as follows: D 1 J 1 = 21 ; J 2 = 22 ; J 3 = 23 ; J 4 = D 2 ; J 5 = 31 B 1 B 2 ; D 3 B 3

7 E 1 = Y. Benveniste, G.W. Milton / J. Mech. Phys. Solids 51 (2003) E 4 = u 1;1 u 1;2 u 1;3 ;1 ;2 ;3 ; E 2 = ; E 5 = u 2;1 u 2;2 u 2;3 ;1 ;2 ;3 ; E 3 = u 3;1 u 3;2 u 3;3 ; : (2.7) Note that, by their above denition, the J-elds are divergence free and the E-elds are curl free. Under the denitions in Eq. (2.7), Eq. (2.6) can be written in the notation of (2.1) as: J i = L ij E j ; E j = U ;j with ; =1; 2; 3; 4; 5 i; j =1; 2; 3 (2.8) where, U 1 = u 1 ; U 2 = u 2 ; U 3 = u 3 ; U 4 = ; U 5 = and L ij has been dened as: L ij = C ij ; ;=1; 2; 3 i; j =1; 2; 3; L ij4 = e ji ; L ij5 = q ji ; =1; 2; 3; i; j =1; 2; 3; L i4j = e ij ; L i5j = q ij ; =1; 2; 3 i; j =1; 2; 3; L i4j4 = ij ; L i5j5 = ij ; i;j =1; 2; 3; L i4j5 = ij ; i;j =1; 2; 3: (2.9) Thus, the following symmetry properties for L ij become apparent in the case of piezomagnetoelectricity: L ij = L ij = L ij = L ji ; ;=1; 2; 3 i; j =1; 2; 3; L ij4 = L ij4 = L j4i = L j4i ; L ij5 = L ij5 = L j5i = L j5i ; =1; 2; 3 i; j =1; 2; 3; L i4j4 = L j4i4 ; j =1; 2; 3; L i4j5 = L j5i4 ; L i5j5 = L j5i5 ; j =1; 2; 3: (2.10) A similar notation was introduced by Barnett and Lothe (1975) in piezoelectricity and by Alshits et al. (1992) in piezomagnetoelectricity. We note here that the piezomagnetoelectric law can be written by using various combinations of the elds ij ;E i ; H i ; ij ;D i ;B i. If one uses the triplet ( ij ;E i ;H i ) on one side of the constitutive equation and the triplet ( ij ;D i ;B i ) on the other side of it, then the tensor entering in the constitutive law will be symmetric and positive denite. On a similar basis, one can therefore represent general linear n-coupled eld phenomena by: J i = L ij E j ; E j = U ;j ; J i;i =0; ;=1; 2;:::;n; i; j =1; 2; 3 (2.11) where U are the potentials, and J i are the uxes. For cases in which elasticity is part of the treated coupled eld phenomena, the displacements are considered as potentials and the corresponding stress vectors of (2.7) are considered as uxes. In these circumstances we will choose to refer to the elasticity component by the indices ; =1; 2; 3. For a discussion of the representation of elasticity as a special case of

8 1780 Y. Benveniste, G.W. Milton / J. Mech. Phys. Solids 51 (2003) coupled eld phenomena, see Section 6.4 in TC. Finally, we mention here that in (2.11) rotation of the coordinates act directly on space indices such as i and j, and may mix some of the eld indices according to the problem being studied. For example, in the case of elasticity rotations act on the eld indices as if they were space indices, but for thermoelectricity rotations do not act at all on the eld indices. 3. General derivation for a coated confocal ellipsoidal inhomogeneityand the related composite ellipsoid assemblages Consider a coated ellipsoidal inhomogeneity with the core and coating properties described in the general setting of (2.11) byl (1) ij ;L(2) ij, respectively. Embed this inhomogeneity in an external medium with properties denoted by L ij. Let the external medium be subjected on its boundary S to a potential eld: U (S)=A kx k ; x S (3.1) where A k are constants. We ask whether there exists a possible relation between the properties L (1) ij ;L(2) ij ;L ij, the geometrical parameters of the coated ellipsoid, and the relative magnitudes of the A k coecients describing the conguration of the external loading so that the potential eld in the outside medium is given throughout by the same form as (3.1): U (x)=a kx k : (3.2) An answer to this question in the case of uncoupled elasticity and for the circumstance in which both the core and coating are isotropic has been given in Section 7.9 of TC. As expected from the work of Grabovsky and Kohn (1995), that analysis was based solely on potential solutions of the uncoupled conductivity problem described in 7.7 of TC. Motivated by the analysis in that section, we proceed here in a general manner so as to allow for anisotropic and coupled behaviour both in the core and coating. Let us introduce ellipsoidal coordinates ; ; which are dened implicitly (see for example, Kellogg, 1929) as the solution to the set of equations x 2 1 c x2 2 c x2 3 c = 1 (confocal ellipsoids); x 2 1 c x2 2 c x2 3 c = 1 (hyperboloids of one sheet); x1 2 c x2 2 c x2 3 = 1 (hyperboloids of two sheets); (3.3) + c 2 3 subject to the restrictions c 2 1 c2 2 c2 3, where c 1; c 2, and c 3 are xed positive constants which determine the coordinate system. The inner surface of the coating will be denoted by c and the outer surface by e. Thus 6 c applies to the core, c 6 6 e to the coating, and e to the outside medium.

9 Y. Benveniste, G.W. Milton / J. Mech. Phys. Solids 51 (2003) Let us assume the following potentials in the core U (1) = A k x k ; (3.4) where the A k are constants. The uxes in the core are given by J (1) i = L (1) ij A j, and full trivially J (1) i;i = 0. In the coating let U (2) = A k x k + a k k = A k x k + a k! ;k ; (3.5) where with k = x k g(c ) 2 c d ; (3.6) (ck 2 + ) (c1 2 + )(c2 2 + )(c2 3 + ) g()=(c )(c )(c ); (3.7) and! satises!; ijj =0;!; jj =1. The form of (3.5) (3.7) is motivated by the uncoupled conductivity solution for the same geometry, and the fact that k =!; k is shown, for example, in Section 7.9 of TC. It will be now shown that, for given constituent properties L (1) ij ;L(2) ij of the core and coating, certain solutions can be found for the constants a k ;A k ;A k. These will allow the determination of the uxes Ji, and potentials U in the outside medium which, in their turn, will imply certain constraints on the possible properties L ij. Under these conditions, when the outside medium is subjected on its boundary S to (3.1), the eld in it will be given by (3.2). It turns out that the equilibrium of the uxes in the coating will allow a solution for the coecients a k if certain constraints are fullled among the properties of the coating L (2) ij. Once an admissible solution for these coecients is found, then the interface conditions at the core-coating interface will allow the determination of the constants A k. The interface conditions at the coating-external medium interface, on the other hand, will provide the A k coecients (and thus the U potentials), as well as the uxes Ji, in the outside medium. The uxes in the coating are given by J (2) i = L (2) ij A j + H ijk!; jk ; H ijk = L (2) ij a k; (3.8) and their equilibrium necessitates J (2) i;i =0 H ijk!; ijk =0: (3.9) Making use of the nature of L (2) ijk in (2.11), and the denition of the coecients a k in Eq. (3.5), and assuming that elasticity is part of L (2) ijk when =1; 2; 3, it can be established that the H ijk are the coecients of a fourth order tensor when =1; 2; 3, and the coecients of a third order tensor when =4; 5;:::;n. We will now show that the last equation in (3.9) imposes a restriction on H ijk, and thus on L (2) ij and a k. Dene the functions p ijk =!; ijk. Clearly, these functions remain the same if we interchange any of the indices and also satisfy the constraint p ijj =0. In the two-dimensional case, for example, p 111 and p 222 can be taken as independent

10 1782 Y. Benveniste, G.W. Milton / J. Mech. Phys. Solids 51 (2003) functions (strictly speaking, they are not independent since p 111;222 =p 222;111 ; however, for the purpose of our analysis, they can be treated as independent functions). In terms of them p 122 = p 111 ;p 211 = p 222, and of course one can permute the indices. In the three-dimensional case we can chose the functions p 123 ;p 122 ;p 133 ;p 211 ;p 233 ;p 311, and p 322 to be our seven independent functions. In terms of these there is: p 111 = p 122 p 133 ; p 222 = p 211 p 233 ; p 333 = p 311 p 322 : (3.10) Since the chosen seven p ijk are independent, their coecients in the expansion of (3.9) need to vanish. This yields seven constraints among the components of H ijk. For simplicity of notation, denoting H ijk by H () ijk, we get: H () 333 = H () H () H () 113 ; H() 333 = H () H () H () 223 ; H () 222 = H () H () H () 112 ; H() 222 = H () H () H () 332 ; H () 111 = H () H () H () 221 ; H() 111 = H () H () H () 331 ; H () H () H () H () H () H () 312 =0: (3.11) The above equations constitute 7n constraints among the 3n coecients a k and the elements of L (2) ij. Clearly, (3.11) will be satised if H takes the form: H () ijk = H ijk = V i jk + W j ik + Z k ij + R ijk + S jik + T kij ; (3.12) where jk denotes the Kronecker delta and R ijk ;S jik ;T kij are antisymmetric in their last pair of indices. Conversely, it can be shown that if (3.11) holds then H can necessarily be expressed in the form of (3.12). We omit here the proof since the representation (3.12) is not strictly needed in our analysis, even though it does allow one to establish some consistency properties of H ijk as shown below. Stating that the structure of H ijk needs to be of the form (3.12) is an alternative way of expressing the constraints (3.11) on a general H ijk. Since the H ijk transform as the coecients of a fourth order or as the coecients of a third order tensor, it can be readily established that after a rotation of coordinates the structure of H ijk, as expressed in (3.12), remains the same (and from this analysis it follows that the V i, W i, and Z i can each be assumed to transform as the coecients of a second order tensor for =1; 2; 3 and as the coecients of a vector for =4; 5;:::). Thus, the constraints (3.11) onl (2) ijk and a k are frame independent. This is expected since these constraints arise from the condition H ijk p ijk =0 which holds for all third order tensors satisfying p ijj =0, and this condition is obviously frame independent. Consequently, if (3.11) is satised for some choice of L (2) and a, it will also be satised if we rotate the tensor of the coating material and rotate a accordingly (without having to rotate the axes of the coated ellipsoid). In order to proceed at this stage, we assume that under certain possible constraints on L (2) ij, there exist some solutions for the coecients a k. The determination of the coecients A k is then achieved by considering the following conditions at the interface = c : U (1) =c = U (2) =c ; J (1) i n i = J (2) =c i n i ; (3.13) =c

11 Y. Benveniste, G.W. Milton / J. Mech. Phys. Solids 51 (2003) where n denotes the unit normal to the core-coating interface. In view of (3.4) (3.6), the rst condition in (3.13) is automatically satised. In order to implement the second condition in (3.13), it is rst noted from (3.5) to(3.7) that as c, one k (x) j 2(ck 2 + ) j j = 2x j =(c 2 j + ) x 2 1 (c ) + x 2 2 (c ) + x 2 3 (c ) : (3.14) On the other hand, making use of n i = x i =(ci 2 + ) ; (3.15) (c1 2 + ) (c2 + ) + (c3 2 + ) x1 2 x2 2 x3 2 it readily follows that as c one k (x)=@x j n k n j, i.e.!; jk n j n k. Thus it turns that the second equation in (3.13) will be satised if the coecients A k fulll the following equations: n i L (1) ij A j = n i L (2) ij A j + n i H ijk n j n k : (3.16) Using the constraint conditions (3.11) (or the representation (3.12)) it can be shown that: n i H ijk n j n k = n i H () iii ; (3.17) and consequently (3.16) will be satised if (L (1) ij L(2) ij )A j = h i (3.18) holds, where h i = V i + W i + Z i = H iii : (no sum on i) (3.19) From the tensorial nature of L ijk and A k (as implied by (3.5)), it is noted that (3.18) demands that the h i behave under rotation as the coecients of as a second order tensor if =1; 2; 3 and as the coecients of a vector for each value of =4; 5;:::;n. In fact, the tensorial nature of H ijk, together with the representation (3.12) allows one to prove that the expression V i + W i + Z i = h i does transform as second order tensor when =1; 2; 3, and a vector when =4; 5;:::;n. Eqs. (3.18) are a set of 3n equations for the 3n coecients of A k. Unless L (1) L (2) has some degeneracy there will be a unique solution to this equation. Note that if elasticity is part of L ijk then this tensor is symmetric in the rst two indices when =1; 2; 3 and consequently L (1) L (2) is degenerate. Thus, rstly, in view of the structure of (3.18) and (3.19) it follows that we have the additional requirement that H () jjj needs to be symmetric with respect to interchange of j and when =1; 2; 3.

12 1784 Y. Benveniste, G.W. Milton / J. Mech. Phys. Solids 51 (2003) This gives the following new constraints on the H tensor in addition to those present in (3.11): H (1) 222 = H (2) 111 ; H(1) 333 = H (3) 111 ; H(2) 333 = H (3) 222 : (3.20) Secondly, in this case the system of Eqs. (3.18) becomes linearly dependent, unless of course we eliminate from this system the three equations which repeat themselves under an interchange of i and for =1; 2; 3. On the other hand, it is readily veried that for k and =1; 2; 3, the A k coecients appear always in the combination of A k + A k in (3.18) (this is expected since one can always add a rigid body rotation to A k without aecting the strains). Thus, for =1; 2; 3, the unknowns in these equations can now be identied as the strains (A k + A k )=2 and not as the A k. Proceeding in this manner, it is seen that the number of unknowns is now reduced by three and the system in (3.18), with the above mentioned three equations eliminated, can now generally be solved. The next step is the determination of the coecients A k for the potentials in the outside medium together with the uxes Ji. The conditions at the interface = e are: U (2) =e = U =e ; J (2) m i = J =e im i =e ; (3.21) i where m denotes the unit normal to the coating-external medium interface. Use of (3.2), (3.5), (3.8) in(3.21) provide after some manipulations: A k = A k + a k g k (no sum on k); (3.22) g k = 1 e g(c ) 2 c d ; (3.23) (ck 2 + ) (c1 2 + )+(c2 2 + )(c2 3 + ) Ji = L (2) ij A j + H ijk jk g k + f 1 h i = L (2) ij A j + f 1 h i ; (3.24) where we have indicated by f 1 the volume fraction of the core, dened as the ratio of its volume to the total volume of the inhomogeneity, and have used the identity g(c ) f 1 = g(e ) : (3.25) The constants g 1 ;g 2 ;g 3 satisfy g 1 + g 2 + g 3 = f 2 ; g 1 0; g 2 0; g 3 0; (3.26) and conversely for any choice of the numbers g 1 ;g 2 ;g 3 satisfying these constraints there is a unique corresponding coated confocal ellipsoid (Tartar, 1985). Finally it is also observed that if elasticity is part of L, then for =1; 2; 3 and k, the A k coecients will also always appear in (3.24) in the form A k + A k. With A k, and J i thus obtained, the external medium has to possess a tensor L which fullls J i = L ija j: (3.27)

13 Y. Benveniste, G.W. Milton / J. Mech. Phys. Solids 51 (2003) Since the tensors A, H and h depend linearly on the tensor a it is evident that (3.27) can be rewritten in the form P ik a k = L ijq jk a k ; (3.28) where the tensors P and Q depend on L (1) ; L (2) ;f 1 ;g 1 ;g 2 ;g 3 as implied by Eqs. (3.18), (3.22), and (3.24). For a given choice of L (2), Eqs. (3.11) (and (3.20) when applicable) place linear constraints on the tensor a and restrict it to lie in a subspace. Let m be the dimension of this subspace. Then, provided the tensor (L (1) L (2) ) 1 exists so that P and Q are well dened, Eqs. (3.28) determine the action of the tensor L on an m-dimensional subspace. In summary, we have proved the following theorem: Theorem 1. A coated confocal ellipsoid of arbitrary eccentricity consisting of a core with tensor L (1) and coating with tensor L (2) can be inserted into an appropriately chosen homogeneous medium without disturbing the surrounding appropriately chosen linear potentials in it, provided L (2) is such that H (as dened by (3.8)) satises (3.11) for at least one choice of the coecients a k and provided L (1) is such that L (1) L (2) is non-singular. The potentials (3.2) in the surrounding homogeneous medium must be chosen so that the coecients A k satisfy (3.22) where the A k are the solutions of (3.18), h i is given by (3.19), and the coated ellipsoid shape factors g k are given by (3.23). The tensor L of the surrounding medium must be chosen so that (3.27) is satised where the uxes Ji are given by (3.24) in which f 1 is the volume fraction of the core in the coated ellipsoid. If elasticity elds with potentials indexed by =1; 2; 3 are present, then L (1) L (2) is necessarily singular. Yet, for k, and =1; 2; 3, the A k coecients now appear in (3.18) and (3.24) in the combination (A k + A k ) which factor out rigid body motions and this combination can be uniquely determined from (3.18), as long as there is no further degeneracy in L (1) L (2). The tensor L (2) must now be such that H satises both (3.11) and (3.20) for at least one choice of the coecients a k. Such inhomogeneities which can be inserted into a homogeneous material without disturbing the surrounding linear potentials are called neutral inhomogeneities, see, for example, Chapter 7 in TC. It therefore becomes clear that one can continue to add such inhomogeneities (with the same orientation but possibly scaled in size) to the host medium without disturbing the external eld (3.2). In fact, the host medium can be lled up completely by such inhomogeneities (see the Appendix). This is called a composite ellipsoid assemblage. Its eective tensor L ij needs to obey (3.28). Only in cases where the a k coecients can be fully arbitrary, does Eq. (3.28) allow the determination of the complete eective tensor L. It is noted from (3.22), (3.18), (3.19) and the second in (3.8) that the ability of constructing a solution for an arbitrary a k implies the feasibility of having a solution for a fully arbitrary uniform external loading represented by the A k coecients. We will show that this happens in the case of conductivity with an isotropic coating (which may however involve the Hall eect) and n-coupled eld phenomena with scalar potentials like magnetoelectricity provided the coating is isotropic. In the remainder of the paper we apply the general analysis

14 1786 Y. Benveniste, G.W. Milton / J. Mech. Phys. Solids 51 (2003) of this section to ellipsoid composite assemblages in the setting of various physical phenomena. 4. Conductivityand related phenomena Consider uncoupled eld phenomena governed by a curl free eld E, and a divergence free eld J: J(x)=(x)E(x); J =0; E =0: (4.1) This system represents several physical phenomena: electrical conductivity, dielectrics, magnetism, thermal conduction, diusion, ow in porous media. It is convenient to represent J and E by 3 1 column matrices, and bya3 3 square matrix which may be non-symmetric so as to allow magnetotransport and convection enhanced diusion phenomena, see Section 2.2 in TC. We will consider real only. Since there is a single eld in question, we let ; = 1, and identify the elements of L ij of the general notation in (2.11) as follows: L 1111 = 11 ; L 2121 = 22 ; L 3131 = 33 ; L 1121 = L 2111 = 12 ; L 1131 = L 3111 = 13 ; L 2131 = L 3121 = 23 : (4.2) Furthermore, the coecients a k ;A k ;A k will be represented in the present case by (a 1 ;a 2 ;a 3 ); (A 1 ;A 2 ;A 3 ); (A 1 ;A 2 ;A 3 ) and correspond to = 1. Some algebra shows that the constraint conditions (3.11) now become: (2) 33 a 3 =( (2) 31 + (2) 13 )a 1 + (2) 11 a 3; (2) 33 a 3 =( (2) 32 + (2) 23 )a 2 + (2) 22 a 3; (2) 22 a 2 =( (2) 21 + (2) 12 )a 1 + (2) 11 a 2; (2) 22 a 2 =( (2) 23 + (2) 32 )a 3 + (2) 33 a 2; (2) 11 a 1 =( (2) 12 + (2) 21 )a 2 + (2) 22 a 1; (2) 11 a 1 =( (2) 13 + (2) 31 )a 3 + (2) 33 a 1; ( (2) 12 + (2) 21 )a 3 +( (2) 32 + (2) 23 )a 1 +( (2) 13 + (2) 31 )a 2 =0: (4.3) The rst and sixth equality in (4.3) can be written as [ ( (2) 31 + (2) 13 ) ((2) 33 (2) 11 ) ]( ) ( ) a1 0 = ; (4.4) ( (2) 33 (2) 11 ) ((2) 31 + (2) 13 ) a 3 0 the second and fourth equality as [ ( (2) 23 + (2) 32 ) ((2) 33 (2) 22 ) ]( ) ( ) a2 0 = ; (4.5) ( (2) 33 (2) 22 ) ((2) 23 + (2) 32 ) a 3 0 and the third and fth equality as [ ( (2) 21 + (2) 12 ) ((2) 22 (2) 11 ) ]( ) ( ) a1 0 = : (4.6) ( (2) 22 (2) 11 ) ((2) 12 + (2) 21 ) a 2 0

15 Y. Benveniste, G.W. Milton / J. Mech. Phys. Solids 51 (2003) Since the determinant of these matrices is the sum of two squares, it is clear that the only way (4.4) (4.6) can be satised is if (2) 11 = (2) 22 = (2) 33 ; (2) 21 = (2) 12 ; (2) 31 = (2) 13 ; (2) 32 = (2) 23 ; (4.7) under which, one can have completely arbitrary (a 1 ;a 2 ;a 3 ). The conditions in (4.7) allow also the fulllment of the seventh constraint in (4.3). Imposing (4.7) in the square matrix of (4.1), we rewrite the (2) matrix in the form: (2) = (2) 11 (2) (2) 12 (2) 12 (2) (2) 23 (2) 13 (2) 23 (2) 11 : (4.8) This allows the treatment of electrical conductivity in the presence of a magnetic eld where the o-diagonal terms reect the inuence of the Hall coecient. Strictly speaking, if a magnetic eld H acts on an isotropic material then in addition to the Hall eect, the symmetric part of the conductivity tensor will become anisotropic due to magnetoresistance. However, whereas the Hall eect occurs at rst order in H, the magnetoresistive eects occur at second order in H, and thus may be neglected to rst order in H when H is small. Eqs. (3.18), (3.22), and (3.24) in the present case can be shown to reduce to: ( (1) (2) )A = (2) 11 a; A = A + Ga; J = (2) A + f 1 (2) 11 a; (4.9) where a = a 1 a 2 a 3 ; A = A 1 A 2 A 3 ; A = A 1 A 2 A 3 ; G = g g g 3 : (4.10) Since A; A, and J depend linearly on the fully arbitrary components of a, use of (3.27) in the form J = A, together with (4.9) shows after some manipulation that the tensor assumes the following closed form representation: = (2) + f 1 (2) 11 [((1) (2) ) 1 (2) 11 + G] 1 : (4.11) This is the exact expression for the eective conductivity of the corresponding composite ellipsoidal assemblage consisting of aligned coated ellipsoids. In this equation all tensors are referred to a Cartesian coordinate system whose axes are parallel to the principal axes of a prototype coated ellipsoid (in other coordinate systems G is non-diagonal and the eigenvectors of G are aligned with the principal axes of the prototype coated ellipsoid: G transforms as a second-order tensor). One can check that (4.11) is consistent with the result of Stroud and Bergman (1984) which states that when (x) has a constant anti-symmetric tensor added to it then the effective tensor will have the same tensor added to it. If the o-diagonal terms of (2) vanish and (1) is symmetric then (4.11) reduces to (8.8) in TC if one

16 1788 Y. Benveniste, G.W. Milton / J. Mech. Phys. Solids 51 (2003) notes that d c1 0 0 G = f 2 M; M =(D c f 1 D e )=f 2 ; D c = 0 d c2 0 ; 0 0 d c3 d e1 0 0 D e = 0 d e2 0 ; (4.12) 0 0 d e3 where D c and D e are the depolarization tensors of the interior and exterior surfaces of the coating in the inhomogeneity. The tensor M is positive semi-denite with Tr M =1. 5. Elasticity Let us consider the case in which both the core and the coating are orthotropic with the three planes of symmetry coinciding with the symmetry planes of the ellipsoids. An orthotropic material is dened by the following constitutive law: 11 = C C C ; 22 = C C C ; 33 = C C C ; 32 =2C ; 13 =2C ; 12 =2C ; (5.1) where the material constants have been written using the engineering notation which is described later in (7.4) in the more general piezomagnetoelectric context. Using the general notation of (2.11) yields the following representation with ; =1; 2; 3: 11 L L L 1133 u 1; L L u 1; L L u 1; L L u 2;1 22 = L L L 2233 u 2;2 ; L L u 2; L L u 3; L L u 3;2 33 L L L 3333 u 3;3 (5.2)

17 Y. Benveniste, G.W. Milton / J. Mech. Phys. Solids 51 (2003) where the following identication is made: L 1111 = C 11 ; L 2222 = C 22 ; L 3333 = C 33 ; L 1122 = L 2211 = C 12 ; L 1133 = L 3311 = C 13 ; L 2233 = L 3322 = C 23 ; L 3223 = L 3232 = L 2323 = L 2332 = C 44 ; L 3113 = L 3131 = L 1313 = L 1331 = C 55 ; L 2112 = L 2121 = L 1212 = L 1221 = C 66 : (5.3) Writing out the constraint conditions (3.11) and (3.20) it can be proved that with an orthotropic coating having a positive denite tensor C the a k are necessarily diagonal. Let us denote for simplicity the non-vanishing a 11 ;a 22 ;a 33 coecients by a 1 ;a 2 ;a 3. After identifying the non-vanishing components of H ijk, under the specic choice of the material parameters (5.3) and the a k coecients, the constraint conditions (3.11) can be shown to become: 33 a 3 = 13 a (a 1 + a 3 ); 33 a 3 = 23 a (a 2 + a 3 ); 22 a 2 = 12 a (a 1 + a 2 ); 22 a 2 = 23 a (a 2 + a 3 ); 11 a 1 = 12 a (a 1 + a 2 ); 11 a 1 = 13 a (a 1 + a 3 ); (5.4) whereas the symmetry conditions (3.20) now become automatically fullled. Some manipulation of (5.4) shows that six of the nine elastic constants can be written in terms of (a 1 ;a 2 ;a 3 ) and the remaining three constants 11 ;C(2) 22 ;C(2) 33 ; as follows: 12 = a 1 11 a 2 22 ; 13 (a 2 a 1 ) = a 1 11 a 3 33 ; 23 (a 3 a 1 ) = a 2 22 a 3 33 ; (a 3 a 2 ) 66 = a2 2 C(2) 22 a2 1 C(2) 11 (a 2 2 a2 1 ) ; 55 = a2 1 C(2) 11 a2 3 C(2) 33 (a 2 1 a2 3 ) ; 44 = a2 3 C(2) 33 a2 2 C(2) 22 (a 2 3 a2 2 ) : (5.5) By this process one can arbitrarily assign the coecients a 1 ;a 2 ;a 3 together with 11 0; C(2) 22 0; C(2) 33 0, and generate the remaining material parameters. One has to check that the resulting elasticity matrix is of course positive denite. Note that for an isotropic coating the choice of a 1 = a 2 = a 3 always provides a solution in (5.4). Thus, for the orthotropic case a positive denite solution is always ensured if the resulting elasticity matrix is close to being isotropic, corresponding to a 1 a 2 a 3. Next, we proceed to the determination of the A k ;A k ;J i coecients for a xed choice of a 1 ;a 2 ;a 3 and satisfying (5.5). An examination of (3.18) and (3.22) for the present case in which both the core and coating are orthotropic shows that the elements of A k and A k with k remain undetermined but obey A k = A k ; A k = A k and represent a rigid body rotation. They can thus be ignored. We denote the diagonal elements of A k by A k = A kk ; A k = A kk (no sum on k). Eqs. (3.18), (3.22),

18 1790 Y. Benveniste, G.W. Milton / J. Mech. Phys. Solids 51 (2003) and (3.24) reduce to: ( C (1) )A = D a; A = A + Ga; J = A + f 1 D a; (5.6) where G has been dened by (4.10) and (3.23), the column vectors a; A; A have the same conguration as in (4.10), and the following matrices have been introduced: 11 C (r) 11 C (r) 12 C (r) 13 J = 22 ; C (r) = C (r) 12 C (r) 22 C (r) 23 r =1; 2; 33 C (r) 13 C (r) 23 C (r) D = : (5.7) Now consider a composite ellipsoid assemblage consisting of coated ellipsoids of all sizes with the same aspect ratios and with their principle axes being aligned with the coordinate axes. It follows from (3.27) that a subset of the eective tensor C of the assemblage (which may be fully anisotropic depending on the packing mode) must obey the following relations: 11 = C11A 11 + C12A 22 + C13A 33; 22 = C12A 11 + C22A 22 + C23A 33; 33 = C 13A 11 + C 23A 22 + C 33A 33; C 14A 11 + C 24A 22 + C 34A 33 =0; C 15A 11 + C 25A 22 + C 35A 33 =0; C16A 11 + C26A 22 + C36A 33 =0: (5.8) To summarize, if the moduli of the coating are such that (5.4) is satised for some choice of a 1 ;a 2, and a 3, then the eective moduli Cij of the assemblage must satisfy (5.8), in which ij and A ij are found by substituting (4.10) and (5.7) in(5.6), solving the rst equation in (5.6) for A, and then substituting the result in the second and third equations in (5.6). Some special cases of the above analysis are now discussed. If one lets the coating be cubic ( 11 = C(2) 22 = C(2) 33 ; C(2) 12 = C(2) 13 = C(2) 23 ; C(2) 44 = C(2) 55 = C(2) 66 ) it is readily veried from conditions (5.4) that one should have either 11 =2C(2) 66 + C(2) 12 or C(2) 11 =. The former case renders the coating isotropic, whereas the latter leads to a C tensor which is not positive denite. For the case of an isotropic coating (with the choice of a 1 = a 2 = a 3 ) and an isotropic (or cubic) core one gets from the rst of (5.6) A 11 = A 22 = A 33 which corresponds to a state of strain in the core which is hydrostatic. This recovers the result in Section 7.9 of TC. Next, let us discuss the possibilities of having a transversely isotropic coating. For transverse isotropy about the x 3 axis, we have: 44 ; C(2) 12 = C(2) 44 C 11 = k + G T ; C 12 = k G T ; C 13 = C 23 = l; C 33 = n; C 44 = C 55 = G L ; C 66 = G T ; (5.9)

19 Y. Benveniste, G.W. Milton / J. Mech. Phys. Solids 51 (2003) where k is the plane strain bulk modulus for lateral dilatation without axial straining; n is the modulus in uniaxial straining in the x 3 -direction, l denotes the cross-modulus in the same test, and G L and G T are the longitudinal and transverse shear modulus respectively. It can readily be shown that the only possible solution is obtained under: (l (2) + G (2) L )2 +(G (2) L n (2) )(k (2) + G (2) T G (2) L )=0; a 1 = a 2 ; a 3 = l(2) + G (2) L n (2) G (2) a 1 ; (5.10) L and of course under the provision that the positive deniteness conditions G (2) T 0; G(2) L 0; k(2) 0; n (2) 0; (l (2) ) 2 k (2) n (2) (5.11) are fullled. If the coated inhomogeneity is spheroidal (g 1 = g 2 ), with both the coating and core being transversely isotropic, and the packing of the microgeometry is such that the eective behaviour is transversely isotropic, then one can show that equations (5.6) and (5.8) result in the following two exact relations for the subset of three eective moduli k ;l ;n : where 2(q 0 + g 1 )(k k (2) )+(p 0 + g 3 m 0 )(l l (2) )=f 1 (k (2) + G (2) T ); (5.12) 2(q 0 + g 1 )(l l (2) )+(p 0 + g 3 m 0 )(n n (2) )=f 1 n (2) m 0 ; (5.13) q 0 =[(n (1) n (2) )(k (2) + G (2) T ) (l(1) l (2) )n (2) m 0 ]=d; p 0 =2[ (l (1) l (2) )(k (2) + G (2) T )+(k(1) k (2) )n (2) m 0 ]=d; d = 2[(k (1) k (2) )(n (1) n (2) ) (l (1) l (2) ) 2 ]; m 0 =(l (2) + G (2) L )=(n(2) G (2) L ): (5.14) If the spheroids become innite cylinders (g 3 0), then one has also Hill s relations between k ;l ;n Hill (1964): k f 1 k (1) f 2 k (2) l f 1 l (1) f 2 l (2) = l f 1 l (1) f 2 l (2) n f 1 n (1) f 2 n (2) = k(1) k (2) l (1) l (2) (5.15) (we note here that relations (5.15) would also result from the derivation in the present paper if one considers, a priori, a microgeometry which is cylindrical, see Section 8 and the remarks following Eq. (8.28)). Finally, it turns out that, with g 3 =0, g 1 = g 2 = f 2 =2, among the four relations in (5.13) and (5.15) only three are independent, and they can be solved to yield the exact expressions for the three moduli

20 1792 Y. Benveniste, G.W. Milton / J. Mech. Phys. Solids 51 (2003) k ;l ;n of the composite cylinder assemblage found by Hashin and Rosen (1964) and Hashin (1972). 6. Coupled-elds with scalar potentials In this section we consider physical phenomena of the thermoelectric type with curl-free elds (E 1 ; E 2 ;:::;E n ) and n divergence free uxes (J 1 ; J 2 ;:::;J n ). We assume that the coating has rotational symmetry with respect to the x 3 axis, referred to as symmetry of Class 6 so that its tensor takes the form: 1 L (2) ij = ijr (j) + e ij3t with r (2) = r(1) ; (6.1) where e ijk is the completely antisymmetric tensor taking the value 1 ( 1) if ijk is an even (odd) permutation of 123 and zero otherwise (note that the superscript (j) on r in (6.1) does not refer to the phases but to the j th component of L (2) ij ). Since Eqs. (3.11) remain invariant if we interchange the indices i and j of H () ijk, it follows that the coecients t do not enter in these equations. With the choice (6.1) ofl (2) ij,itis straightforward to check that Eqs. (3.11) reduce to Da (1) = Da (2) = Da (3) = 0; (6.2) where D is the n n matrix with elements D = r (1) r(3) k =1; 2; 3 a (k) is the n-component vector with elements a (k), and for each value of = a k. This results in only one condition on the elements of L (2), namely that the determinant of D must vanish. Assuming that there are no further accidental degeneracies, so that rank D = n 1, then a (k) = c k V; (6.3) where V is the null vector of D, and the constants c 1 ; c 2 and c 3 can be arbitrary. Thus the tensor a lies in a three-dimensional subspace and accordingly we expect (3.28) to determine the action of L on a three-dimensional subspace. This will give 9n relations among the elements of L, and 9n 3 relations when L is symmetric. Finally it is of interest to study the case of an isotropic coating. In this case the constraint condition det D = 0 is trivially fullled for an arbitrary choice of a k. Eqs. (3.18), (3.22) and (3.24) can be now cast in the form: (L (1) L (2) )Â = L (2) â; Â = Â + Ĝâ; Ĵ = L (2) Â + f 1 L (2) â; (6.4) 1 For a classication of symmetry classes in magnetoelectric media as well as piezoelectric and piezomagnetic solids see the book by Bhagavantam (1966), as well as the earlier book by Nye (1957). Class 6 is a class which exhibits six-fold symmetry about a preferred axis (it can be checked that (6.1) exhibits in, in fact, rotational symmetry for any rotation about the x 3 -axis.)