Elastic Metamaterials with Simultaneously Negative Effective Shear Modulus and Mass Density

We propose a type of elastic metamaterial comprising fluid-solid composite inclusions which can possess a negative shear modulus and negative mass density over a large frequency region. Such a material has the unique property that only transverse waves can propagate with a negative dispersion while longitudinal waves are forbidden. This leads to many interesting phenomena such as negative refraction, which is demonstrated by using a wedge sample and a significant amount of mode conversion from transverse waves to longitudinal waves that cannot occur on the interface of two natural solids.

Figure 1

(a) A schematic figure of the building block of the metamaterial: a rubber-coated water cylinder embedded in a foam host. (b) Effective medium parameters obtained from EMT. Solid (black), dashed (red), short-dashed (blue), and dash-dotted (cyan) curves correspond to effective medium parameters, ${\rho }_e$, ${\kappa }_e$, ${\mu }_e$, and ${\kappa }_e+{\mu }_e$, respectively.

Figure 2

(a) Band structure of a triangular array of rubber-coated water cylinders embedded in foam host as shown in Fig. 1b. (b) The solid curve (blue) is the transmission coefficient for the longitudinal wave incidence and the dashed curve (red) is for the transverse wave incidence along the $\Gamma M$ direction.

Figure 3

(a) Schematic figure of the sample. $i$ and $\theta$ are incident and refracted angles, respectively. $n$ is the surface normal of the interface between the metamaterial and the host. $\Gamma M$ and $\Gamma K$ directions are along horizontal and vertical directions, respectively. The incident wave is coming from the left, shooting normally onto the vertical surface of the wedge. (b) (Left) The equifrequency surface in the $\mathit{k}$ space of the foam (red dashed curve for transverse modes and blue short-dashed curve for longitudinal modes) and the metamaterial (green solid curve) at $\tilde{f}=0.2$. Brillouin zone is depicted by solid back lines. (Right) The enlarged inner part of the equifrequency surface. ${\mathbf{v}}_g$ and $\mathit{k}$ are the group velocity and wave vector inside the metamaterial, respectively. ${\mathbf{k}}_l$ (${\mathbf{v}}_{gl}$) and ${\mathbf{k}}_t$ (${\mathbf{v}}_{gt}$) are longitudinal and transverse wave vectors (group velocities) inside the host, respectively. (c) Total flux distribution for a transverse wave incidence on the wedge. The colors and arrows represent the magnitude and directions, respectively. (d) The far-field total flux distribution for the transverse wave incidence. Solid (black) and dashed (red) curves correspond to longitudinal and transverse waves, respectively.

Figure 4

A schematic graph of $s\mathrm{\text{−}}p$ mode conversion on the interface between a double-negative elastic metamaterial and a normal elastic medium.