We consider the linear-response properties of composites made of two isotropic materials. We concentrate on linear-response phenomena, with nonvanishing cross coefficients, such as the magnetoelectric effect, the thermoelectric effect, and coupled, multispecies diffusion. We find that the moduli of the composite must obey a number of compatibility relations, involving the moduli of the components, but totally independent of the mixture ratio of the components and the microstructure of the composite. The compatibility conditions take the form C(L,$L_a$,$L_b$)==${\mathrm{LL}}_a^{-1}$$L_b$-$L_b$$L_a^{-1}$L=0, where $L_a$,$L_b$ are the response matrices of the components, and L is that of the composite. These come about because there exists a choice of driving forces and fluxes—an eigenbasis—that are decoupled in both components.

When the composite itself is isotropic, these constitute n(n-1)/2 relations that its moduli must obey (for real response matrices), leaving only n independent coefficients to be specified—n being the number of driving fields. When the composite is anisotropic, the elements of L are spatial tensors and the same conditions apply as relations between these tensors. In the latter case one can eliminate the moduli of the components, and obtain relations that the moduli of the composite must satisfy among themselves. The mere fact that the composite is made of two isotropic components, without any further information, imposes strong relations among its moduli. For example, we show that all the tensor moduli of such a composite are symmetric. The same compatibility conditions must also be obeyed by the response matrices of any three composites of the same two isotropic components; so, knowledge of the moduli of two composites supplies constraints on any third, or, for that matter, on the components themselves. We discuss the general properties of the compatibility conditions and their possible relation to parallel and series construction of composites, and demonstrate our findings for the most common case of two driving fields in three dimensions. The compatibility conditions apply not only for the effective response matrices of well-homogenized composites, but also for the local response—at any point within some region that is filled with an arbitrary two-phase mixture—to the potentials on the boundary of the region. All the above apply, with little change, to the case where the response matrices are complex-symmetric or Hermitian. We have nothing to add to the lore on composites, in the case where all the cross coefficients vanish; the compatibility conditions are identities in this case. All our results remain intact for multiphase composites, when all the components but two are either perfect conductors or perfect insulators.

• Received 11 January 1989

DOI:https://doi.org/10.1103/PhysRevA.40.1568