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The acoustic cloaking theory of Norris (2008) permits considerable freedom in choosing the transformation function f from physical to virtual space. The standard process for defining cloak materials is to first define f and then evaluate whether the materials are practically realizable. In this paper, this process is inverted by defining desirable material properties and then deriving the appropriate transformations which guarantee the cloaking effect. Transformations are derived which result in acoustic cloaks with special properties such as 1) constant density 2) constant radial stiffness 3) constant tangential stiffness 4) power-law density 5) power-law radial stiffness 6) power-law tangential stiffness 7) minimal elastic anisotropy., 9 pages, 8 figures

Transformation acoustics uses invertible maps to transform a cloaked region to a region with a smaller scatterer. The “elastic” or Norris class of acoustic cloaking theory is inherently broadband, in that it does not rely on the presence of resonant effects. To achieve the properties required by the transformation theory, we utilize pentamode materials, which admit the use of finite mass but require anisotropic stiffness throughout the material. These types of materials do not exist naturally and must be fabricated through the use of metamaterials. One of the challenges associated with metamaterials for use in dynamic applications is the distribution of mass in a unit cell and its consequences for frequency-dependent behavior in the metamaterial. Various homogenization techniques for recovering wave speed properties of the metamaterial unit cells may predict contradictory wave speed values for pentamodal structures. These disagreements can be explained through the distribution of inertia in the material a...

The field of acoustic metamaterial research has been driven, first, by transformation acoustics cloaking theory. This and other theoretical approaches optimize an acoustic effect through definition of artificial material properties which at present are only achievable using metamaterial technology. Metamaterials, however, present separate challenges in optimization for any desired acoustic effect: in particular, their dynamic behavior depends on parameters unrelated to acoustics, and exhibits wave propagation behavior more complex than elastic or acoustic continua. Homogenization methods which assume Cartesian symmetry are a staple in metamaterial design, but these only approximate a feasible optimal design. Moreover, total reliance on homogenized continuum models provides no information about the actual microstructure performance, and presents problems in functionally graded applications. To overcome this, we augment our Cartesian homogenization processes with high-resolution finite element models to optimize the design. Comparatively computationally expensive implicit FEM is avoided; specifically tailored time-domain wave propagation codes make the analyses feasible. Examples of the process, combining Cartesian homogenization estimates with high-resolution microstructural wave propagation solutions, will be shown.The field of acoustic metamaterial research has been driven, first, by transformation acoustics cloaking theory. This and other theoretical approaches optimize an acoustic effect through definition of artificial material properties which at present are only achievable using metamaterial technology. Metamaterials, however, present separate challenges in optimization for any desired acoustic effect: in particular, their dynamic behavior depends on parameters unrelated to acoustics, and exhibits wave propagation behavior more complex than elastic or acoustic continua. Homogenization methods which assume Cartesian symmetry are a staple in metamaterial design, but these only approximate a feasible optimal design. Moreover, total reliance on homogenized continuum models provides no information about the actual microstructure performance, and presents problems in functionally graded applications. To overcome this, we augment our Cartesian homogenization processes with high-resolution finite element models to opt...

Acoustic metamaterials use sub-wavelength, anisotropic, and inhomogeneous microstructures. Macroscopic properties can be related to the microstructure using homogenization theory Hassani and Hinton [Comput. Struct. 69, 719–738 (1998), which allows an analyst to confirm the extent to which a candidate metamaterial microstructure meets the requirements for a pentamode cloaking material. Norris [Proc. R. Soc. Ser. A, 464, 2411–2434 (2008)] presented a theory of transformation acoustics that enables the realization of inhomogeneous pentamode acoustic materials having anisotropic elastic tensors, isotropic density and finite mass. This theory describes the spatially varying material properties in terms of a mapping, which for separable geometries may be generated using a scalar function. This function, the constraints on its behavior implicit in the Norris theory, and the material equations constitute the defining relations for pentamode transformation acoustics. Previously, analytic work in transformation acoustics developed the material properties after having fixed a transformation. By reversing the process, we create a number of new families of pentamode cloaking materials. We validate the concept with three-dimensional explicit transient finite element simulations.

Norris’ theory of transformation acoustics enables the realization of pentamode acoustic materials having anisotropic density and finite mass. Two additional extensions of the theory are considered, with a view toward demonstrating feasibility for practical applications. First, specializations of the theory developed by Norris [“Acoustic cloaking theory,” Proc. R. Soc. London, Ser. A 464, 2411–2434 (2008)] are considered in order to develop transformations corresponding to pentamode acoustic metamaterials with specified functional forms (constant or powerlaw) for density and stiffness. Ray‐tracing and finite element simulations of wave propagation through such pentamode acoustic media are presented. Next, the design of acoustic materials for objects composed of simple geometric shapes (e.g., a cylinder with spherical endcaps) is considered. It is shown that certain classes of transformations are well‐suited for the design of such shapes and validate the concept with finite element simulations.

Norris [“Acoustic cloaking theory,” Proc. R. Soc. London, Ser. A 464, 2411–2434 (2008)] presented a theory of transformation acoustics that enables the realization of pentamode acoustic materials having anisotropic density and finite mass. The design of acoustic materials for 3‐D objects composed of combinations of simple geometric shapes, e.g., a cylinder with spherical end‐caps, is considered. Certain classes of transformations which are well‐suited for the design of such shapes are described and conditions under which these transformations result in materials which minimize acoustic reflections are proposed. The concept is validated using 3‐D explicit transient finite element simulations, which closely emulate the physics of wave propagation in the proposed materials.