UPPER AND LOWER BOUNDS FOR THE OVERALL PROPERTIES OF A NONLINEAR COMPOSITE DIELECTRIC .1. RANDOM MICROGEOMETRY

The direct extension of the Hashin-Shtrikman methodology to nonlinear composite problems generally produces at most one new bound - either an upper bound or a lower bound - and in some cases produces no new bound at all. This paper is devoted to the construction of bounds, of generalized Hashin-Shtrikman type, for any nonlinear composite whose behaviour can be characterized in terms of a convex potential function. The construction relies on the use of a nonlinear 'comparison medium' and trial fields with the property of 'bounded mean oscillation'. This permits the exercise of control over the size of the penalty incurred from the use of a nonlinear, as opposed to linear, comparison medium. In cases where a linear comparison medium is adequate, the already established bounds of Hashin-Shtrikman type are reproduced. The exposition is presented in the context of bounding the properties of a nonlinear dielectric, for which a single bound was obtained previously by one of the authors. The approach, however, is applicable more generally.