# Recent Advances in Non-Traditional Elastic Wave Manipulation by Macroscopic Artificial Structures
## 2 Elastic Metamaterials
### Elastic Hyperlens
An extention [109] of this idea considered a more-general elastic transformation problem that includes primary (P) and shear vertical (SV) motions, in addition to shear horizontal (SH) motion. The constitutive equations required for elastodynamics cloaking, which is restricted in the form for isotropic elasticity, were obtained; numerical simulations showed that the presence of an elastic cloak has greatly reduced scattering (Figure 9b). An elastic cloak to control the scattering of bending waves in isotropic heterogeneous thin plates has also been proposed [110]. The scattering cancellation technique was applied to the biharmonic operator (\(\
abla^{4}\)) to design the cloak. A fourth-order biharmonic equation with appropriate boundary conditions to describe the propagation of bending waves in thin plates was derived from the  
Figure 9: (**a**) Total displacement field of anti-plane shear waves. Line source is shown as a white circle. Left side: a classic elastic cloak (upper) and a designed pre-stress elastic cloak (lower). Right side: scattering from a cavity of radius \(k_{s}R=\pi/20\) (upper) and \(2\pi\) (lower) in an unstressed medium, where \(k_{s}\) is the shear wavenumber of the medium and \(R\) is the annulus inner radius of the cavity; (**b**) Top: Total displacement field of SH waves without a cloak (left) and with a cloak (right). Center: Scattered displacement field of P waves without a cloak (left) and with a cloak (right). Bottom: Scattered displacement field of SV waves without a cloak (left) and with a cloak (right); (**c**) Displacement field distributions of cloaked (left) and uncloaked (right) rigid object. White arrows: vector direction. Colored contours: amplitude. (**a**) adapted from [108]. (**b**) adapted from [109]. (**c**) adapted from [110]. generalized elasticity theory [111, 112], and yielded the displacement field distributions of an obtained cloak (Figure 9c). A directional cloak that can protect a region from a flexural wave within a range of bandgap frequencies has been demonstrated [113]. Recently, elastic cloaking has been extended to the seismic waves that propagate through the Earth. Seismic MTMs, one of the most important application in EMTMs, are starting with the use of PCs and metasurface [35, 36, 37, 38, 39, 40, 41, 42]. One of the ways to demonstrate elastic cloaking is to realize form-invariant Willis coupling metamaterial including an additional degree of freedom. Since it is first proposed by [PERSON] [114], Willis coupling has been explored in elastodynamics and acoustics analogous to the bianisotropy parameter in electromagnetism. In electromagnetic metamaterials, bianisotropy enables the coupling of magnetic and electric phenomena at the subwavelength scale. In bianisotropic media, the constitutive relations in electromagnetism [115] are written as  
\[\mathbf{D}=\epsilon\mathbf{E}+\mathbf{\xi}\mathbf{H},\quad\mathbf{B}=\mathbf{\mu} \mathbf{H}+\mathbf{\zeta}\mathbf{E}, \tag{9}\]  
where \(\mathbf{D}\) is the electric displacement field, \(\mathbf{E}\) is the electric field, \(\mathbf{B}\) is the magnetic flux density field, \(\mathbf{H}\) is the magnetic field, \(\epsilon\) is the permittivity tensor and \(\mathbf{\mu}\) is the permeability tensor. \(\mathbf{\xi}\) and \(\mathbf{\zeta}\) are coupling tensors. The Willis constitutive relations [114, 116] can be expressed as  
\[ \begin{pmatrix}\langle\mathbf{\sigma}\rangle\\ \langle\mathbf{p}\rangle\end{pmatrix} = \begin{pmatrix}\mathbf{C}_{eff}&\mathbf{ S}_{eff}\\ \mathbf{\hat{S}}_{eff}&\mathbf{\rho}_{eff}\end{pmatrix}\begin{pmatrix}\langle \mathbf{e}\rangle\\ \langle\dot{\mathbf{u}}\rangle\end{pmatrix} , \tag{10}\]  
where \(\mathbf{\hat{S}}_{eff}\) is the adjoint of \(\mathbf{S}_{eff}\), \(\mathbf{e}\) is the strain tensor, \(\mathbf{\sigma}\) is the stress tensor, \(\mathbf{\mathrm{C}}_{eff}\) is the effective stiffness tensor, \(\mathbf{u}\) is the displacement vector, \(\mathbf{p}\) is the momentum density vector and \(\mathbf{\rho}_{eff}\) is the effective mass density tensor.