LINEAR RESPONSE OF GENERAL COMPOSITE SYSTEMS TO MANY COUPLED FIELDS

We treat the linear response of an arbitrary composite system to many coupled driving fields, as in the thermoelectric or magnetoelectric effects. The system is made of any number of, possibly anisotropic, components. The response consists of some component of the fluxes (currents) measured at an arbitrary point in the system and is considered as responding to the boundary conditions. We expound the problem pertaining to such a response and investigate the general properties of the relevant response matrices. Let Lkma be the elements of the p response matrices, La (1ap), characterizing the components. We show that the functional dependence of the response coefficients (scrLij) of the composite on them [scrLij=scrLij(Lkm a)] is subject to two constraints: First, it enjoys certain covariance properties under linear transformations of the fields; for any real regular matrix W we must have W-1scrL(WL1W,...,WLpW)W-1=s crL(L1,...,Lp), where W is the transpose of W. Second, it possesses disjunction properties; when the component response matrices are block diagonal of the same form, so must scrL, and, furthermore, the elements of scrL within each block depend only on Lkmas in the corresponding blocks. From these general requirements follow many applications, of which we describe a few. In particular, the cases of nearly uncoupled phenomena (when the cross-coefficients of all the components are small), and of nearly uniform systems (such as a polycrystal made of a nearly isotropic crystal) afford simple and powerful results. In the former case, we find that the problem breaks into approximately decoupled, two-field problems. In the latter case we show that if La=L0+scrEa, where Ea are small then, to second order in the elements of Ea, one has Lkm=(L0)km+aqakma+1/2) a,bQab(aL0-1b)km, where , qa, and Qab are numbers, independent of L0, of the particular cross-coefficient involved, or of the phenomenon under study; they depend only on the structure of the system and the measurement setup. The general coupled-field response matrix of a nearly uniform composite material is reducible to the single-field case in the second-order approximation: The coefficients qa and Qab can be determined by measurements of single-field properties, such as the conductivity, and used to deduce all the direct- and cross-response properties of the system for any multifield phenomenon. We also comment on higher orders. © 1990 The American Physical Society.