Page 43 - \Winter 2015

!!
𝑟𝑟!
𝑑𝑑𝑉𝑉!= 𝑓𝑓𝑑𝑑𝑑𝑑 𝑟𝑟
!! 〈𝜌𝜌 〉
! !!! ! 𝑟𝑟
𝑓𝑓𝑑𝑑𝑑𝑑
Acoustic cloaking is more achievable than its optical counterpart because of the much larger wavelengths in- volved.
as a pentamode material (PM). An elastic solid is character-
ized by six modes of deformation, a PM is the limiting case
where five of the six are "soft" modes (Milton and Cherkaev,
𝑟𝑟
and the pressure relation 𝐶𝐶𝑝𝑝 = −𝛁𝛁. 𝐯𝐯
Eliminating particle velocity v yields the wave equation
𝛁𝛁. 𝜌𝜌!!𝛁𝛁𝑝𝑝 −𝐶𝐶𝑝𝑝𝐶𝐶=𝑝𝑝0=w−h𝛁𝛁ic.h𝐯𝐯isanisotropicbyvirtueoftheten!so!rρ_1.
𝜌𝜌𝐯𝐯 = −𝛁𝛁𝑝𝑝
〈𝜌𝜌!! 〉
force balance
𝐶𝐶𝑝𝑝= −𝛁𝛁.𝐯𝐯 𝜌𝜌𝐯𝐯= −𝛁𝛁𝑝𝑝
! 𝑟𝑟!
!!!
𝑓𝑓𝑑𝑑𝑑𝑑
arrays of rigid cylinders in water (Torrent and Sánchez-De- hesa, 2008b). The anisotropic wave speed follows from the
𝑑𝑑𝑉𝑉 =
𝑟𝑟
〈𝜌𝜌!!〉
𝜌𝜌𝐯𝐯 = −𝛁𝛁𝑝𝑝
The first papers on acoustic cloaking assumed anisotropic
1995) with one stiff mode. PMs generalize the property of an
𝑑𝑑𝑉𝑉 =
𝛁𝛁.𝜌𝜌 𝛁𝛁𝑝𝑝−𝐶𝐶𝑝𝑝=0
!! acoustic fluid that it can shear without effort but resists hy-
mass density. Cummer and Schurig (2007) noted the anal- ogy between the EM wave equation with anisotropic per- mittivity and the acoustic equation with a density tensor to describe a 2D cylindrically symmetric cloak. Chen and Chan (2007) proposed a spherically symmetric cloak with anisotropic density. Most subsequent acoustic cloaking lit- erature is based on anisotropic inertia, what we call inertial cloaking (IC). Particular realization of ICs are in principle feasible using layers of isotropic fluid (Torrent and Sánchez- Dehesa, 2008a), a strategy which has proved very success- ful in realizing carpet cloaks, as discussed below. However, fully enveloping cloaks require extreme anisotropy near the inner boundary that can only be achieved by alternat- ing layers of fluids with extremely small and large densities. At the same time, the compressibility must be such that the homogenized value is that of TA. The cylindrical or spheri- cal layered cloak does not seem to be possible with existing fluids. Models such as (Torrent and Sánchez-Dehesa, 2008a) require hundreds of fluids with different properties, some with very large compressibility and density. One possible so- lution is to take advantage of the non-uniqueness of TA and find the best possible transformation for a given set of fluids (e.g. 2 or 3 (Norris and Nagy, 2010)) but this also requires that the constituents have widely disparate properties not found in available materials.
Another possibility exists: anisotropic wave speeds can be achieved with anisotropic bulk modulus rather than density. It turns out that TA is fundamentally different from its EM counterpart where the transformation uniquely defines the EM material and, for instance, the tensors of electric permit- tivity and magnetic permeability display the same level of anisotropy for a transformation of the vacuum. In acoustics, by contrast, there is a wide range in material properties that can yield a given transformation. The non-uniqueness comes from the freedom to introduce an arbitrary positive definite symmetric divergence free matrix S into TA (Norris, 2008; Norris and Shuvalov, 2011). The inertial cloak corresponds to S=I, the identity, which partly explains why this degree of freedom in TA had not been noticed earlier. Any other choice of S leads to anisotropic stiffness in the sense of elas- ticity, however it is a special type of elastic material known
𝛁𝛁.𝜌𝜌 𝛁𝛁𝑝𝑝−𝐶𝐶𝑝𝑝=0
drostatic compression with a stress −pI, where ρ is acoustic pressure. The PM stress is proportional to S.
Pentamode cloaks have, in principle, distinct advantages over ICs. For instance, cylindrical or spherical cloaks with isotropic density are possible, in which case the total cloak mass is simply the mass of the original, virtual region (Nor- ris, 2009). In contrast, the mass of a perfect IC becomes unbounded (Norris, 2008). The PM in the cloak must still have continuously varying properties defined by TA. Scan- drett et al. (2010) examined the effect of piecewise layering in a spherical cloak, and found that an optimized three layer PM cloak provides better target strength reduction than a 3-layer IC. The best performance was found by combin- ing both properties in a PMIC cloak. Scandrett and Vieira (2013) showed that the dominant scattering from heavily fluid loaded thin shells in the mid- and high-frequency re- gimes can be essentially eliminated by PM cloaking.
The current limitation on PM cloaking is fabrication. Solid materials with five soft modes and one stiff mode with de- sired stress state S can be achieved with periodic foam-like networks in which the microstructure lattice members only support axial forces. In practice, this means thin members that are flexible in bending but stiff in compression. Fab- rication of such microstructured lattices is possible using rapid prototyping and related technologies; for instance Bückmann et al. (2014) designed and fabricated a PM "un- feelability" cloak, essentially a static cloak for elastic fields. The difficulty as far as cloaking sound in water is concerned is to achieve just the right properties. In order to get density and stiffness values similar to those of water using thin lat- tice members of low volume fraction the structural material must be very dense and stiff relative to water. Metals provide the appropriate reservoir of density and stiffness. The first realization of a metal-based microstructure with 2D PM be- havior close to water, called "metal water" (Norris and Nagy, 2011), was an Aluminum lattice with hexagonal unit cells. The metal lattice model has also been studied by Layman et al. (2013) who simulated a slab of PM designed to provide a 2D acoustic illusion of Figure 1.
| 41