There are sometimes certain tensors T, called translations, such that if T is added to the local tensor field L(x), then the effective tensor will also be shifted by T. We saw a trivial example of such a tensor in section 2.5 on page 30, where it was remarked that if the local specific heat is shifted by a constant –t, the effective specific heat would also be shifted by that same constant [see equations (2.25) and (2.26)]. Here we discuss other translations and show how they lead to useful predictions about effective moduli. We will see later, in section 13.3 on page 274, that such translations are connected with quadratic null Lagrangians, when T is self-adjoint, and, more generally, with weakly continuous bilinear functions when T is not necessarily self-adjoint. There is an extensive literature on the theory of such functions (see section 13.3 on page 274 for some references) and the examples of this section fall under the umbrella of the more general theory. In particular, it is easy to check that all of the constant translations T discussed here satisfy the necessary and sufficient algebraic condition given by Murat (1978), theorem 3; see also Tartar (1979).

Translations applied to conductivity

The duality transformation is clearly a discrete transformation. However, as we will now see, it is just one example of a continuous group of fractional linear transformations, each having the special property that the effective tensor undergoes the same transformation as the local conductivity tensor.