Wave characteristics of extremal elastic materials

The elasticity tensors of extremal materials are rank-deficient, they have inherent easy deformation modes without spending energy. Kinds of materials are classified by the number of vanishing eigenvalues of their elastic matrices, namely from unimode, bimode to pentamode. They may exhibit unprecedented capacity to manipulate waves, as already exemplified by pentamode materials. The peculiar property of these materials on wave propagation lies in reducing the number of slowness surfaces as well as the opening of these surfaces along certain directions determined by easy deformation modes. We demonstrate these wave characteristics according to the classification of extremal materials by analyzing their acoustic tensors. It is shown that the number of slowness surfaces is determined by the number of independent characteristic force vectors provided by the hard modes of the extremal materials on any wave front plane. As a consequence, there is only one slowness surface for pentamode materials, and two for quadramode materials. Trimode materials may have two or three depending whether the three force vectors provided by the materials are coplanar or not. All the other modes definitely have three slowness surfaces. The opening of these surfaces depends on null of these force vectors along certain directions, controlled by the soft deformation modes in these materials. Concrete examples are also given to illustrate these findings. A device of broadband zero-refractive-index for elastic wave is proposed with a trimode material. These works pave the way to design wave devices by exploiting extremal materials.