Elsevier

Wave Motion

Volume 48, Issue 6, September 2011, Pages 525-538
Wave Motion

Elastic cloaking theory

https://doi.org/10.1016/j.wavemoti.2011.03.002Get rights and content

Abstract

Transformation theory is developed for the equations of linear anisotropic elasticity. The transformed equations correspond to non-unique material properties that can be varied for a given transformation by selection of the matrix relating displacements in the two descriptions. This gauge matrix can be chosen to make the transformed density isotropic for any transformation although the stress in the transformed material is not generally symmetric. Symmetric stress is obtained only if the gauge matrix is identical to the transformation matrix, in agreement with Milton et al. [1]. The elastic transformation theory is applied to the case of cylindrical anisotropy. The equations of motion for the transformed material with isotropic density are expressed in Stroh format, suitable for modeling cylindrical elastic cloaking. It is shown that there is a preferred approximate material with symmetric stress that could be a useful candidate for making cylindrical elastic cloaking devices.

Research Highlights

► It is shown that the equations of linear anisotropic elasticity can be transformed into either material equations with symmetric or nonsymmetric stress. ► Symmetric stress is obtained only if a quantity relating displacement before and after transformation, the gauge matrix, is chosen to be equal to the transformation matrix. ► It is also shown that there is a preferred approximate material with symmetric stress that could be a useful candidate for making cylindrical elastic cloaking devices.

Introduction

Interest in cloaking of wave motion has surged with the demonstration of the possibility of practical electromagnetic wave cloaking [2]. The principle underlying the effect is the so-called transformation or change-of-variables method [3], [4] in which the material parameters in the physical domain are defined by a spatial transformation. The concept of material properties defined by transformation is not restricted to electromagnetism, and has stimulated interest in applying the same method to other wave fields. The first applications in acoustics were obtained by direct analogy with the electromagnetic case [5], [6], [7]. It was quickly realized that the fundamental mathematical identity behind the acoustic transformation is the equivalence [8] of the Laplacian in the original coordinates to a differential operator in the transformed (physical) coordinates, according to Div Grad f  J div J 1FFt grad f, where F = x / X is the deformation gradient of the transformation (see Section 2) and J = det F. This connection, plus the realization that the tensor within the operator can be interpreted as a tensor of inertia means that the homogeneous acoustic wave equation can be mapped to the equation for an inhomogeneous fluid with anisotropic density.

However, the material properties for acoustic cloaking do not have to be identified as a fluid with a single bulk modulus and a tensorial inertia. There is a large degree of freedom in the choice of the cloaking material properties [9], [10]: a compressible fluid with anisotropic density is a special case of pentamode materials [11], [12] with anisotropic inertia. The non-uniqueness of the material properties (for a given transformation) is a feature not found in the electromagnetic case, where, for instance, the tensors of electric permittivity and magnetic permeability are necessarily proportionate for a transformation of the vacuum. The extra freedom in the acoustic case means that either or both of the scalar parameters, density and elastic stiffness (bulk modulus), can become tensorial quantities after the transformation. While most papers on acoustic cloaking have considered materials with scalar stiffness and tensorial inertia, e.g. [5], [6], [7], [13], [14], [15], [16], [17], [18], see [19] for a review, there is no physical reason for such restricted material properties. Cloaking with such materials requires very large total mass [9], [20], but the more general class of pentamode materials with anisotropic inertia do not have this constraint. In fact, it is often possible to choose the material properties so that the inertia is isotropic, in which case the total mass is simply the mass of the equivalent undeformed region [10]. This property can be realized if the transformation is a pure stretch, as is the case when there is radial symmetry. This distinction between the cloaking material properties is critical but, judging by the continued emphasis on anisotropic inertia in the literature, does not seem to have been fully appreciated. Apart from [9], [10] there have been few studies [21], [22] of acoustic cloaking with anisotropic stiffness.

The non-uniqueness of the transformed material properties found in the acoustic theory transfers to elastodynamics. The first study of the transformation of the elastodynamic equations by Milton et al. [1] concluded that the appropriate class of constitutive equations for the transformed material is the Willis equations for material response. The general form of the Willis equations are [1], [23]divσ=p˙,σ=Ceff*e+Seff*u,p=Seff*e+ρeff*u˙,where e=12(u+(u)t), * denotes time convolution and is the adjoint. The stress in Eq. (1b) is symmetric, and the elastic moduli enjoy all of the symmetries for normal elasticity, viz. Cklijeff = Cijkleff and Cjikleff = Cijkleff. Brun et al. [24] considered the transformation of isotropic elasticity in cylindrical coordinates for the particular transformation function used by [3], [4] and found that the transformed material properties are those of a material with isotropic inertia and elastic behavior of Cosserat type. The governing equations for Cosserat elastic materials [25] aredivσ=ρeffu¨,σ=Ceffu,with elastic moduli satisfying the major symmetry Cklijeff = Cijkleff but not the minor symmetry, Cjikleff  Cijkleff. This implies that the stress is not necessarily symmetric, σt  σ, and that it depends not only on the strain e (the symmetric part of the displacement gradient) but also upon the local rotation ω=12(u(u)t). Not only are the parameters such as Ceff in Eqs. (1a)–(1c) and (2a)–(2b) different, the constitutive theories are mutually incompatible: one has symmetric stress, the other a non-symmetric stress. We show in this paper that both theories are possible versions of the transformed elastodynamic equations, and that they are only two from a spectrum of possible constitutive theories. Apart from the two references mentioned [1], [24], the only other example of transformation elasticity concerns flexural waves obeying the biharmonic equation [26], which is beyond the realm of the present paper.

The purpose here is to consider the transformation method for elastodynamics, and to describe the range of constitutive theories possible. The starting point is the observation [10] that the extra degrees of freedom noted for the acoustic transformation can be ascribed to the linear relation between the displacement fields in the two coordinate systems. This “gauge” transformation introduces a second matrix or tensor, in addition to the deformation gradient from the change of coordinates. As discussed in [10], the variety of acoustically transformed material properties arises from the freedom in the displacement gauge. The same freedom is also present in the elastic case, and as we will show, it allows one to derive a broader class of constitutive properties than those suggested by Milton et al. [1] and by Brun et al. [24]. The material properties found in these studies correspond to specific choices of the gauge matrix.

Cloaking is achieved with transformations that deform a region in such a way that the mapping is one-to-one everywhere except at a single point, which is mapped into the cloak inner boundary. This is a singular transformation, and in practice, the mapped region would be of finite size, e.g. a small sphere, for which the mapping is everywhere regular. Our objective here is to understand the nature of the material necessary to produce the transformation effect, in particular, what type of constitutive behavior is necessary: such as isotropic or anisotropic inertia.

The outline of the paper is as follows. The notation and setup of the problem are given in Section 2 where the displacement gauge matrix is introduced. The general form of the transformed equations of motion are presented in Section 3. Constitutive equations resulting from specific forms of the gauge matrix are discussed in Section 4, particularly the Willis equations and Cosserat elasticity, which are shown to coincide under certain circumstances. The special case of transformed acoustic materials is discussed in Section 5. The elastic transformation theory with isotropic density is applied in Section 6 to radial transformation of cylindrically anisotropic solids. Based on this formulation, an elastic material with isotropic density and standard stress–strain relations is proposed in Section 6.3 as an approximation to the transformed material. A summary of the main results is given in Section 7.

Section snippets

Notation and setup

Two related configurations are considered: the original Ω, and the transformed region ω, also called the physical or current domain. The transformation from Ω to ω is described by the point-wise deformation from X ∈ Ω to xω. The symbols ∇, ∇X and div, Div indicate the gradient and divergence operators in x and X, respectively, and the superscript t denotes transpose. The component form of div E is ∂Ei / xi or ∂Eij / xi when E is a vector and a second order tensor-like quantity, respectively.

General form of the transformed equations

Under the transformation and the gauge change the energy density transforms as E0E=W+T according to EdV=E0dV0, so thatE=W+T=12J1{CIJKL(0)(ujAjJ),i(ulAlL),kFiIFkK+ρ0u˙iu˙jAiIAjI}.

Hence,W=12J1CIJKL(0)FiIFkK(ujAjJ),i(ulAlL),k,T=12u˙tρu˙,where the (symmetric) density tensor isρ=ρt=ρ0J1AAt,

The equations of motion in the deformed, or current material, are determined by the Euler–Lagrange equations of the Lagrangian density L=WT, asAjJ(J1CIJKL(0)FiI(ulAlL),kFkK),iρiju¨i=0.

Using the identity(J1F

Willis equations: A = F

The absence of the minor symmetries under the interchange of i and j in Cijkleff and Sijleff of Eqs. (15a)–(15c) implies that the stress is generally not symmetric. Symmetric stress is guaranteed if QijIJ=QjiIJ (see Eq. (13)), which occurs if the gauge matrix is of the form A = ζF, for any scalar ζ  0, which may be set to unity with no loss in generality. This recovers the results of Milton et al. [1] that the transformed material is of the Willis form, Eqs. (1a)–(1c). As noted in [1], this is the

General form of transformed equations

We consider the simpler but special case of acoustics in order to further understand the structure of the elastic transformation theory. The acoustic equations are unique in the sense that they are the single example of a pentamode material [11], [12] commonly encountered in mechanics. We will demonstrate that the pentamode property introduces a unique degree of freedom not available in the fully elastic situation.

The elastic stiffness of an acoustic fluid isC0=K0II,which is of pentamode form

Applications in cylindrical elasticity

The non-uniqueness in the form of the transformed equations of elasticity provides the designer of elastic cloaking devices with a wide variety of possible materials from which to choose. These range from materials with Willis constitutive behavior in Eq. (1a)–(1c), to Cosserat materials in Eq. (25), each special cases of the general constitutive relations (14a)–(14b) to (15a)–(15c). The same non-uniqueness exist for acoustic cloaking, where designs based on inertial cloaking on the one hand,

Discussion

A complete transformation theory has been developed for elasticity. The material properties after transformation of the elastodynamic equations are given by Eqs. (13) through (15). The constitutive parameters depend on both the transformation and gauge matrices, F and A, and do not necessarily have symmetric stress. It was shown in Section 4 that a priori requiring stress to be symmetric implies that the material must be of Willis form (1), with A = F as Milton et al. [1] found. The emphasis here

References (32)

  • G.W. Milton et al.

    On cloaking for elasticity and physical equations with a transformation invariant form

    New J. Phys.

    (2006)
  • D. Schurig et al.

    Metamaterial electromagnetic cloak at microwave frequencies

    Science

    (2006)
  • A. Greenleaf et al.

    On nonuniqueness for Calderon's inverse problem

    Math. Res. Lett.

    (2003)
  • J.B. Pendry et al.

    Controlling electromagnetic fields

    Science

    (2006)
  • S.A. Cummer et al.

    One path to acoustic cloaking

    New J. Phys.

    (2007)
  • H. Chen et al.

    Acoustic cloaking in three dimensions using acoustic metamaterials

    Appl. Phys. Lett.

    (2007)
  • S.A. Cummer et al.

    Scattering theory derivation of a 3D acoustic cloaking shell

    Phys. Rev. Lett.

    (2008)
  • A. Greenleaf et al.

    Full-wave invisibility of active devices at all frequencies

    Commun. Math. Phys.

    (2007)
  • A.N. Norris

    Acoustic cloaking theory

    Proc. R. Soc. A

    (2008)
  • A.N. Norris

    Acoustic metafluids

    J. Acoust. Soc. Am.

    (2009)
  • G.W. Milton et al.

    Which elasticity tensors are realizable?

    J. Eng. Mater. Technol.

    (1995)
  • G.W. Milton

    The theory of composites

    (2001)
  • L.-W. Cai et al.

    Analysis of Cummer–Schurig acoustic cloaking

    New J. Phys.

    (2007)
  • S.A. Cummer et al.

    Material parameters and vector scaling in transformation acoustics

    New J. Phys.

    (2008)
  • Y. Cheng et al.

    A multilayer structured acoustic cloak with homogeneous isotropic materials

    Appl. Phys. Lett.

    (2008)
  • A.N. Norris

    Acoustic cloaking in 2D and 3D using finite mass

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