Three-point bounds for the overall properties of a nonlinear composite dielectric

The Hashin-Shtrikman methodology for nonlinear composite problemsrelies on the use of a comparison medium and in many of theexamples studied so far the comparison medium has been takento be homogeneous. A related approach originated by P. PonteCastaeda employs a comparison medium which is itself a linearcomposite with the same microgeometry as the nonlinear composite.When the method is applicable, the bounds for the nonlinearproblem then involve bounds for the energy of the linear comparisoncomposite which could include, for example, three-point informationabout the microstructure of the composite. It is, however, onlypossible to obtain at most one bound using a linear comparisonmaterial. A recent approach involves the use of a nonlinearcomparison medium and trial fields with the property of boundedmean oscillation. In this paper the approach is extended byusing a nonlinear comparison composite so that both upper andlower bounds can be obtained which incorporate three-point information.The development is in the context of bounding the propertiesof nonlinear dielectric composites, although it has wider application.