3-POINT BOUNDS FOR THE OVERALL PROPERTIES OF A NONLINEAR COMPOSITE DIELECTRIC

The Hashin-Shtrikman methodology for nonlinear composite problems reli es on the use of a comparison medium and in many of the examples studi ed so far the comparison medium has been taken to be homogeneous. A re lated approach originated by P. Ponte Castaneda employs a comparison m edium which is itself a linear composite with the same microgeometry a s the nonlinear composite. When the method is applicable, the bounds f or the nonlinear problem then involve bounds for the energy of the lin ear comparison composite which could include, for example, three-point information about the microstructure of the composite. It is, however , only possible to obtain at most one bound using a linear comparison material. A recent approach involves the use of a nonlinear comparison medium and trial fields with the property of bounded mean oscillation . In this paper the approach is extended by using a nonlinear comparis on composite so that both upper and lower bounds can be obtained which incorporate three-point information. The development is in the contex t of bounding the properties of nonlinear dielectric composites, altho ugh it has wider application.