This webinar is organised by the European Association of Mechanical, Acoustic and Thermal Metamaterials https://meta-mat.org.
It is now slightly more than 40 years since David Bergman and myself obtained bounds on the effective complex dielectric constant of isotropic composites of two isotropic phases mixed in given proportions. This constant governs the quasistatic response to electromagnetic waves of fixed frequency. Hence it also controls the absorption and refraction of radiation by these composites. There has been renewed interest in these old bounds, in part because they also give shape independent bounds in quasistatics on the amount of energy dielectric or metallic particles of given volume can absorb. Our bounds on the effective complex dielectric constant consisted of a lens shaped region in the complex plane bounded by circular arcs. We show that one bound can be tightened and that the corresponding circular arc corresponds to an assemblage of doubly coated spheres. New hierarchical laminate geometries that attain additional points on the remaining arc are described. Consequently, we see that this bound is very tight, almost optimal. We also bound the quasistatic response to applied electromagnetic fields, or antiplane elastic fields, that can have any variation in time, not necessarily a fixed frequency one. Curiously, judicious choices of the applied field can directly yield the volume fractions of the phases from measurements at specific times. We show how applied fields can be tailored to have this property. The work is joint with my wonderful and energetic collaborators: Christian Kern, Ornella Mattei, Owen Miller, and Mihai Putinar.