Fundamental Principles for Generalized Willis Metamaterials

Metamaterials that exhibit a constitutive coupling between their momentum and strain, show promise in wave manipulation for engineering purposes and are called Willis materials. They were discovered using an effective-medium theory, showing that their response is nonlocal in space and time. Recently, we generalized this theory to account for piezoelectricity, and demonstrated that the effective momentum can depend constitutively on the electric field, thereby enlarging the design space for metamaterials. Here, we develop the mathematical restrictions on the effective properties of such generalized Willis materials, owing to passivity, reciprocity, and causality. The establishment of these restrictions is of fundamental significance, as they test the validity of theoretical and experimental results - and applicational importance, since they provide elementary bounds for the maximal response that potential devices may achieve.