# Recent Advances in Non-Traditional Elastic Wave Manipulation by Macroscopic Artificial Structures
## 2 Elastic Metamaterials
### Elastic Hyperlens
The propagation of a time-harmonic elastodynamic wave equation [99] is written as  
\[\
abla\cdot\mathbf{\sigma}=-\omega^{2}\rho\mathbf{u},\ \ \ \mathbf{\sigma}=\mathbf{C}\
abla \mathbf{u}, \tag{7}\]  
where \(\rho\) is the scalar density, \(\mathbf{u}\) is the displacement vector, \(\mathbf{C}\) is a rank-4 elasticity tensor, \(\mathbf{\sigma}(\mathbf{x})\) is the stress field. Consider a mapping from an initial space to an arbitrary curvilinear space \(\mathbf{x}\rightarrow\mathbf{x}^{\prime}\) and \(\mathbf{u}(\mathbf{x})\)\(\rightarrow\mathbf{u}^{\prime}(\mathbf{x})=(\mathbf{A}^{T})^{-1}\mathbf{u}( \mathbf{x})\) where \(A_{ij}=\frac{\partial\mathbf{x}_{i}^{\prime}}{\partial\mathbf{x}_{j}}\). Then the equation becomes  
\[\
abla^{\prime}\cdot\mathbf{\sigma}^{\prime}=\mathbf{D}^{\prime}\
abla^{\prime} \mathbf{u}^{\prime}-\omega^{2}\mathbf{\rho}^{\prime}\mathbf{u}^{\prime},\ \ \ \mathbf{\sigma}^{\prime}=\mathbf{C}^{\prime}\
abla^{\prime}\mathbf{u}^{\prime}+ \mathbf{S}^{\prime}\mathbf{u}^{\prime}, \tag{8}\]  
where \(\mathbf{S}^{\prime}\) and \(\mathbf{D}^{\prime}\) are rank-3 symmetric tensors and \(\mathbf{\rho}^{\prime}\) becomes a rank-2 density tensor. Unlike the [PERSON]'s equation retain its form after conformal mapping (Equations (5) and (6)), it is clear that the equation after conformal mapping (Equation (8)) has a different form from Equation (7) in the elastic regime. Therefore, studies have been made to realize elastic wave cloaking for special cases by using other methods. A cylindrical cloak to control bending waves, one of several forms of elastic wave, can be achieved by a special case in a thin elastic plate [102]; the elasticity tensor can be represented by a diagonal matrix that has two non-vanishing entries. Numerical simulations (Figure 8a) suggest that a heterogeneous orthotropic thin elastic cloak can control the propagation of bending waves, and thereby extend the electromagnetic and acoustic cloaking mechanisms [103]; in corresponding experiments [104], the designed sample of elastic cloak (Figure 8b) performed well in the frequency range of 200 to 450 Hz (Figure 8c). Figure 8: **(a)** Left: displacement field scattered by an obstacle for a plane wave of frequency 60 Hz (upper), 150 Hz (lower). Right: displacement field scattered by obstacle surrounded by cloak layers for a plane wave of frequency 60 Hz (upper), 150 Hz (lower); **(b)** Designed elastic cloak before filling with PDMS; **(c)** Displacement field for plane waves with frequencies of 200, 300, 400 and 450 Hz from the top. Left column: without cloak and right column: with cloak. Dashed black circles: outer radius of the cloak region; white scale bar: 5 cm. **(a)** adapted from [103]. **(b,c)** adapted from [104]. Another cylindrical cloak for coupled shear and longitudinal waves can be described using a rank-4 asymmetric elasticity tensor [105]. Asymmetric constitutive relations in which only the major symmetry is preserved were found by applying a cloaking transformation to Navier equations for isotropic linear elasticity. The elasticity tensor in the transformed coordinates was not fully symmetric in this study; as a result, the Navier equations remain in shape under this transformation. A generalization [106] of [99] broadens the transformation theory for elasticity. The material parameters of the transformed system were explicitly derived; they depend on both the transformation and gauge matrices, and do not necessarily have symmetric stress. Cloaking of objects from anti-plane elastic waves can be achieved by using nonlinear elastic pre-stress [107]; this approach does not require inhomogeneous anisotropic shear moduli and densities. Nonlinear pre-stress has been proposed as a method to generate elastic cloaks [108]; a finite cloak for anti-plane elastic waves was designed by applying nonlinear pre-stress to an incompressible neo-Hookean hyper-elastic material; this method cloaked the anti-plane elastic waves (Figure 9a).