Coupled multifield linear response of two-phase composites: Exact universal relations

The linear-response matrix, L, of a composite made of two isotropic components must obey a number of compatibility relations, involving the response matrices, $L_a$ and $L_b$, of the components, but totally independent of the mixture ratio of the components and the microstructure of the composite: ${\mathrm{LL}}_a^{\mathrm{-}1}$ $L_b$-$L_b$ $L_a^{\mathrm{-}1}$L=0. For an anisotropic composite, the tensor elements of L must satisfy the same conditions; in this case we obtain relations that the moduli of the composite satisfy among themselves. The mere fact that the composite is made of two isotropic components imposes relations among its moduli.