Translation and related bounds for the response of a nonlinear composite conductor

Lower bounds for the overall energy functions of a class of nonlinear composite conductors are established by two different methods: the translation method, and the use of a comparison linear composite. When the properties of the comparison linear composite are themselves estimated by the translation method, the resulting comparison bound is demonstrated never to be superior to the bound obtained by direct application of the translation method to the nonlinear composite. Examples show, however, that the bounds obtained by these two methods in fact often coincide; when they do, the use of the comparison medium permits the development of more refined bounds, by exploiting better bounds for the linear composite. Particular emphasis is placed on a limiting case of material behaviour for which explicit results can be obtained: a composite dielectric that displays breakdown when the local electric field exceeds a critical value. The formalism delivers an upper bound for the value of the mean field in the composite which induces breakdown in the composite. Bounds are also established for a composite with a power-law relation between current and electric field.