Cloaking has been a main topic of research since the discovery of MTMs. Initial work started in transformation optics [

96,

97] inspired by earlier results [

98]. The characteristic that obscures an object by using electromagnetic waves is the invariance of Maxwell’s equations which are as follows. Time-harmonic Maxwell’s equations [

99] are written as

where

$\omega $ is an angular frequency,

$\mathit{\mu}$(x) is a magnetic permeability, and

$\mathit{\epsilon}$(x) is an electrical permittivity. Consider a mapping transformation

**x**′ =

**x**′(x),

**E**′(

**x**′) = (

**A**${}^{T}$)

${}^{-1}$**E**(

**x**), and

**H**′(

**x**′) = (

**A**${}^{T}$)

${}^{-1}$**H**(

**x**) where matrix

**A** have elements

${A}_{kj}=\frac{\partial {x}_{k}^{\prime}}{\partial {x}_{j}}$. Then the Maxwell’s equation transforms to

where

$\mathit{\mu}$′(

$\mathbf{x}$′) =

**A**$\mathit{\mu}$(

**x**)

**A**${}^{T}$/det

**A** and

$\mathit{\epsilon}$′(

**x**′) =

**A**$\mathit{\epsilon}$(

**x**)

**A**${}^{T}$/det

**A**. By comparing the Equations (

5) and (

6), it can be seen clearly that Maxwell’s equations still retains their form. Most studies of transformation optics consider electromagnetic waves, but this basic principle applies equally to other types of waves, as long as the wave equation remains invariant under coordinate transformations. Therefore, cloaking has also been applied in acoustics [

14,

100,

101]. However, the equations of motions for elastic medium do not guarantee their form-invariant and are mapped to a more general system with non-scalar density according to geometric changes [

99], so cloaking of the elastic acoustic waves has been difficult to achieve with conventional methods. The propagation of a time-harmonic elastodynamic wave equation [

99] is written as

where

$\rho $ is the scalar density,

**u** is the displacement vector,

**C** is a rank-4 elasticity tensor,

$\mathit{\sigma}$(

$\mathbf{x}$) is the stress field. Consider a mapping from an initial space to an arbitrary curvilinear space

$\mathbf{x}$→

**x**′ and

$\mathbf{u}\left(\mathbf{x}\right)$→

**u**′(

**x**) = (

**A**${}^{T}$)

${}^{-1}$**u**(

**x**) where

${A}_{ij}=\frac{\partial {\mathbf{x}}_{i}^{{}^{\prime}}}{\partial {\mathbf{x}}_{j}}$. Then the equation becomes

where

${\mathbf{S}}^{\prime}$ and

${\mathbf{D}}^{\prime}$ are rank-3 symmetric tensors and

${\mathit{\rho}}^{\prime}$ becomes a rank-2 density tensor. Unlike the Maxwell’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).

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.

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 (

${\nabla}^{4}$) 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 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 Willis [

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

where

**D** is the electric displacement field,

**E** is the electric field,

**B** is the magnetic flux density field,

**H** is the magnetic field,

$\mathit{\epsilon}$ is the permittivity tensor and

$\mathit{\mu}$ is the permeability tensor.

$\mathit{\xi}$ and

$\mathit{\zeta}$ are coupling tensors. The Willis constitutive relations [

114,

116] can be expressed as

where

${\tilde{\mathbf{S}}}_{eff}$ is the adjoint of

**S**${}_{eff}$,

**e** is the strain tensor,

$\mathit{\sigma}$ is the stress tensor,

**C**${}_{eff}$ is the effective stiffness tensor,

**u** is the displacement vector,

**p** is the momentum density vector and

${\rho}_{eff}$ is the effective mass density tensor. The overdot denotes a time derivative and < > denotes the ensemble average. Note that

**S**${}_{eff}$ and

${\tilde{\mathbf{S}}}_{eff}$ are called Willis coupling tensors. Willis coupling requires coupling between strain and momentum fields or stress and velocity fields (Equation (

10)). Several approximations and simplifications to the general approach of Willis media have been published [

117,

118,

119,

120,

121,

122,

123]. In fact, most studies of Willis coupling are more active in the acoustics [

124,

125,

126,

127,

128,

129,

130] than in elastodynamics, but these developments in acoustics indicate that Willis metamaterial for elastic waves in solid may become practical. Recently, Willis metamaterial for flexural waves has been reported [

131]. Based on a cantilever bending resonance, effective bianisotropy in a Willis metamaterial for flexural waves was designed and experimentally demonstrated.