We explore the possibilities coming from transformation acoustics and beyond for creating rigid elastic composite shells capable of suppressing the total scattering cross-section of acoustically large objects. The reported design methodology is based on generalized shape and topology optimization, and the outcomes are suitable for rapid prototyping techniques.U.S. Office of Naval Researc

Smith, David R.

Starr, Anthony F.

Urzhumov, Yaroslav A.

DukeSpace

 PHONONICS 2013: 2nd International Conference on Phononic Crystals/Metamaterials, Phonon Transport and Optomechanics 
Sharm El-Sheikh, Egypt, June 2-June 7, 2013 
PHONONICS-2013-0049 
 
Structurally Rigid Elastic Composites for Acoustic Imag-
ing Countermeasures 
Yaroslav A. Urzhumov1 , Anthony F. Starr2, David R. Smith1 
1 Center for Metamaterials and Integrated Plasmonics, Duke University, 130 Hudson Hall, Box 90271 Durham, 
North Carolina, 27708 USA,  
yaroslav.urzhumov@duke.edu 
2 SensorMetrix, 10171 Pacific Mesa Blvd, Ste 305, San Diego, USA 
astarr@sensormetrix.com 
 
Abstract: We explore the possibilities coming from transformation acoustics and beyond for 
creating rigid elastic composite shells capable of suppressing the total scattering cross-section 
of acoustically large objects. The reported design methodology is based on generalized shape 
and topology optimization, and the outcomes are suitable for rapid prototyping techniques. 
Acoustic cloaking has been a subject of interest to the phononics community since the advent of transformation 
acoustics (TA). TA methodology is based on the form-invariance of the scalar Helmholtz equation describing 
acoustic pressure waves in fluids. Much of the past effort in TA was focused on the so-called acoustic metaflu-
ids 1,2,3, which are materials with vanishingly small shear modulus. The acoustics of metafluids is mathematical-
ly almost as simple as acoustics of regular fluids, with the only complication that the density is anisotropic and 
described by a rank-2 tensor. This explains why virtually all recent effort (with a few exceptions) in transfor-
mation acoustics was devoted to metafluids1,2,3.                                
Unfortunately, the structural properties of metafluids are not particularly suitable for deployable acoustic devic-
es. Fluids (and metafluids) are unable to support any amount of shear stress: an infinitesimal amount of shear 
stress leads to a finite deformation. This presents a fundamental difficulty in integrating them with any on-board 
devices mounted on air vehicles, or even on stationary platforms that are subject to wind or streaming water. 
This lack of structural robustness was overcome in the proof-of-concept demonstrations involving acoustic met-
afluids by restricting device geometries to two-dimensional (in-plane) propagation. These geometries allowed 
the use of metafluids formed by arrays of rigid inclusions (rods, bars) which are mechanically disconnected 
from each other and embedded in a homogeneous fluid (air, water, etc.) By construction, such a metamaterial 
geometry does not support in-plane shear wave propagation and therefore acts as a two-dimensional metafluid. 
The difficulty with generalizing this strategy into three dimensions is so fundamental that no demonstration of a 
non-trivially three-dimensional transformation acoustical device has been offered to date, even as merely a 
proof-of-concept experiment. The need for structural integrity and rigidity calls for the development of elastic 
metamaterials with a suitably large shear modulus. Such media would be capable of tolerating finite stress in 
arbitrary directions, whose amount is limited only by their yield or fracture properties. 
Pentamode elastic media. Unlike the scalar Helmholtz equation, the elastic wave equation, at least in its con-
ventional form, is generally not form-invariant4,5, which makes coordinate transformation techniques like TA 
difficult to apply. However, if one restricts the class of elastic materials to a very small subset, known as pen-
tamode media, the elastic wave equation reduces to a single, scalar Helmholtz wave equation. A pentamode 
medium is defined as an elastic medium whose elasticity tensor C, expressed as a 6-dimensional matrix, can be 
factorized as follows: 
  𝐶𝑖𝑗 = 𝐾𝑄𝑖𝑄𝑗 ,        (1) 
where K is scalar elastic coefficient, and Q is a constant 6-dimensional vector. Substitution of this ansatz into 
the elastic wave equation, 
𝜌?̈?𝑖 = ∇ ∙ 𝜎,        (2) 
where 𝜎 = 𝐶 𝜀 is the stress tensor and 𝜀 = ∇𝑇𝑢 is the strain tensor, u being the displacement vector, gives a sca-
lar equation 
𝜌𝐾−1?̈? = ∇2𝑝        (3) 
for the effective scalar pressure 𝑝 = −𝐾𝑄𝑇𝜀. In the frequency domain, ?̈? = −𝜔2𝑝, and the usual scalar Helm-
holtz equation is retrieved. Allowing the scalar modulus K as well as the density tensor ρ to be functions of co-
ordinates (but not the pentamode vector Q), one obtains an equation with inhomogeneous coefficients that is 
 PHONONICS 2013: 2nd International Conference on Phononic Crystals/Metamaterials, Phonon Transport and Optomechanics 
Sharm El-Sheikh, Egypt, June 2-June 7, 2013 
PHONONICS-2013-0049 
 
form-invariant with respect to arbitrary three-dimensional coordinate transformations. The recipes developed for 
acoustic metafluid-based TA devices should therefore be implementable with pentamode elastic media as well. 
Non-pentamode approaches to shear wave management. From the less mathematical angle, the physical rea-
son for the difficulties with general elastic media is co-existence of vastly different types of waves: longitudinal 
pressure waves, or p-waves, and transverse shear waves (s-waves), which co-propagate and generally couple 
with each other. This issue associated with acoustics of elastic media can be referred to as the multimode propa-
gation problem. The coupling occurs at the sharp interfaces between different media, including the fluid/elastic 
medium boundary. In gradient-index structures, the coupling may also occur due to both gradients of effective 
density and its anisotropy, as was shown in Ref.6. Density gradients and anisotropy arise naturally in most 
Transformation Acoustics scenarios. Although gradients and anisotropy can both be mitigated by optimizing the 
choice of coordinate transformations, complete elimination of anisotropy and/or density gradients is usually 
impossible without degrading the performance of resulting devices, or losing versatility in the choices of device 
shapes. 
The strategies described above – metafluids and pentamode media 
– can be all seen as approaches to manage multimode coupling on 
the microscopic level, by means of adjusting the material proper-
ties and/or the metamaterial unit cell architectures. Though such 
strategies, if proven successful, would be widely applicable to 
virtually all conceivable TA devices, it is worthwhile to consider 
more narrowly focused strategies based on macroscopic manage-
ment of the multimode issues. Recently, Urzhumov et al. demon-
strated in Ref.9 that, under certain resonant conditions, one of the 
co-propagating modes (shear waves) can be effectively eliminated 
from the picture at discrete wavelengths, a concept that opens the 
door to TA devices composed of non-pentamode elastic media 
with substantial shear modulus.  
In this work, we demonstrate macroscopic management of shear 
wave-related scattering in a directional acoustic cloak composed 
of an ABS plastic or another elastic material with a Poisson ratio about 0.4 (Figure 1). It is shown that, by means 
of shape optimization of the air voids, scattering cancellation can be achieved for virtually any desirable Poisson 
ratio and in a wide range of wavelength-to-diameter ratios. Composite structures with strongly reduced scatter-
ing cross-section can be designed in a variety of shapes; cylindrical and spherical ones have the advantage of 
reduced design time due to the availability of efficient full-wave axisymmetric Helmholtz solvers.7 
 
References 
                                                     
1 X. Hu, K.-M. Ho, C. T. Chan and J. Zi, “Homogenization of acoustic metamaterials of Helmholtz resonators in fluid”, 
Phys. Rev. B 77, 172301 (2008). 
2 B.-I. Popa and S. Cummer, “Design and characterization of broadband acoustic composite metamaterials”, Phys. Rev. B 80, 
174303 (2009). 
3 T. P. Martin, M. Nicholas, G. J. Orris, L.-W. Cai, D. Torrent et al., “Sonic gradient index lens for aqueous applications”, 
Appl. Phys. Lett. 97, 113503 (2010). 
4 G. W. Milton, “New metamaterials with macroscopic behavior outside that of continuum elastodynamics”, New J. Phys. 9, 
359 (2007). 
5 G. W. Milton, M. Briane and J. R. Willis, “On cloaking for elasticity and physical equations with a transformation invariant 
form”, New J. Phys. 8, 248 (2006). 
6 Y. Urzhumov, F. Ghezzo, J. Hunt and D. R. Smith, “Acoustic cloaking transformations from attainable material proper-
ties”, New J. Phys. 12, 073014 (2010). 
7 Y. Urzhumov, N. Landy and D. R. Smith, “Isotropic-medium three-dimensional cloaks for acoustic and electromagnetic 
waves”, J. Appl. Phys. 111, 123106 92012). 
 
Figure 1. Left: Structurally rigid direc-
tional acoustic cloak fabricated from a 
hard plastic by Fused Deposition Model-
ing. Right: Full-wave acousto-
elastodynamic simulation of a spherical-
shape cloak in water. Color shows pres-
sure amplitude. 


Structurally Rigid Elastic Composites for Acoustic Imaging Countermeasures

http://dukespace.lib.duke.edu/dspace/bitstream/handle/10161/7578/PHONONICS-2013-0049.pdf?sequence=1

Structurally Rigid Elastic Composites for Acoustic Imaging Countermeasures

Abstract

Similar works

Full text

Available Versions

DukeSpace