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.

Figure 1

The schematics of the cross-couplings reported by Pernas-Salomón and Shmuel [52] in composites with elasticity that is intrinsically coupled with other physics, such as piezoelectric and piezomagnetic materials.

Figure 2

The body $\mathrm{\Omega }$ is composed of different piezoelectric materials, the constitutive response of which is given by Eq. (3), as illustrated at the top of the sketch. Effectively, the response of the body is nonlocal with additional cross-couplings, as given by Eq. (7).

Figure 3

The schematics of the reciprocity principle. The source distributions of problems 1 (cyan) and 2 (gray) are illustrated by darts. The resultant rates of the displacement and electric potential fields of problems 1 (blue) and 2 (black) are illustrated by arrows. The body is reciprocal if the power that distribution 1 produces together with the fields excited by distribution 2 (bottom left sketch) is equal to the power that distribution 2 produces together with the fields excited by distribution 1 (bottom right sketch).