Modelling the effective conductivity function of an arbitrary two–dimensional polycrystal using sequential laminates
The effective conductivity tensor σ*of a two-dimensional polycrystalline material depends on the conductivity tensor σ0of the pure crystal from which the polycrystal is constructed and on the geometrical configuration of grains in the polycrystal, represented by a rotation fieldR(x) giving the orientation of the crystal at each pointx.Here it is established that the dependence of σ*on σ0in any polycrystal, with R (x) held fixed, can be mimicked exactly by a polycrystal constructed by sequential lamination. It is first shown that the effective conductivity function is perturbed only slightly if we truncate the Hilbert space of fields in the polycrystal to a finitedimensional space. Then the structure of this finite-dimensional space of fields is shown to be isomorphic to the structure of the finite-dimensional space of fields associated with the sequential laminate. In particular, there is an operation which corresponds to peeling away the layers in the sequential laminate and successively reducing the dimension of the space of fields.