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

The Hashin-Shtrikman methodology for nonlinear composite problems relies on the use of a comparison medium and in many of the examples studied so far the comparison medium has been taken to be homogeneous. A related approach originated by P. Ponte Castaneda employs a comparison medium which is itself a linear composite with the same microgeometry as the nonlinear composite. When the method is applicable, the bounds for the nonlinear problem then involve bounds for the energy of the linear 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 comparison composite so that both upper and lower bounds can be obtained which incorporate three-point information. The development is in the context of bounding the properties of nonlinear dielectric composites, although it has wider application.