MODELING THE EFFECTIVE CONDUCTIVITY FUNCTION OF AN ARBITRARY 2-DIMENSIONAL POLYCRYSTAL USING SEQUENTIAL LAMINATES

The effective conductivity tensor sigma of a two-dimensional polycrys talline material depends on The conductivity tensor sigma0 of the pure crystal from which the polycrystal is constructed and on the geometri cal configuration of grains in the polycrystal, represented by a rotat ion field R(x) giving the orientation of the crystal at each point x. Here it is established that the dependence of sigma on sigma0 in any polycrystal, with R(x) held fixed, can be mimicked exactly by a polycr ystal constructed by sequential lamination. It is first shown that the effective conductivity function is perturbed only slightly if we trun cate the Hilbert space of fields in the polycrystal to a finite-dimens ional space. Then the structure of the finite-dimensional space of fie lds is shown to be isomorphic to the structure of the finite-dimension al space of fields associated with the sequential laminate. In particu lar, there is an operation which corresponds to peeling away the layer s in the sequential laminate and successively reducing the dimension o f the space of fields.