Antiplane elastic wave cloaking using metamaterials, homogenization and hyperelasticity

Abstract We consider the problem of how to cloak objects from antiplane elastic waves using two alternative techniques. The first is the use of a layered metamaterial in the spirit of the work of Torrent and Sanchez-Dehesa (2008) who considered acoustic cloaks, motivated by homogenization theories, whilst the second is the use of a hyperelastic cloak in the spirit of the work of Parnell et al. (2012). We extend the hyperelastic cloaking theory to the case of a Mooney–Rivlin material since this is often considered to be a more realistic constitutive model of rubber-like media than the neo-Hookean case studied by Parnell et al. (2012), certainly at the deformations required to produce a significant cloaking effect. Although not perfect, the Mooney–Rivlin material appears to be a reasonable hyperelastic cloak. This is clearly encouraging for applications. We quantify the effectiveness of the various cloaks considered by plotting the scattering cross section as a function of frequency, noting that this would be zero for a perfect cloak.

[1]  Alexander B. Movchan,et al.  Achieving control of in-plane elastic waves , 2008, 0812.0912.

[2]  William J. Parnell,et al.  Nonlinear pre-stress for cloaking from antiplane elastic waves , 2012, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[3]  Source amplitudes for active exterior cloaking , 2012, 1207.4801.

[4]  Graeme W Milton,et al.  On modifications of Newton's second law and linear continuum elastodynamics , 2007, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[5]  T. Shearer,et al.  Waves in Nonlinear Elastic Media with Inhomogeneous Pre-Stress , 2013 .

[6]  R. Rivlin.  LARGE ELASTIC DEFORMATIONS OF ISOTROPIC MATERIALS. I. FUNDAMENTAL CONCEPTS , 1997 .

[7]  M. Mooney.  A Theory of Large Elastic Deformation , 1940 .

[8]  J. Willis,et al.  On cloaking for elasticity and physical equations with a transformation invariant form , 2006 .

[9]  J. Willis.  Dynamics of composites , 1997 .

[10]  T. Lewis,et al.  Scattering of elastic waves by a cylindrical cavity in a solid , 1976 .

[11]  E. Cosserat,et al.  Théorie des Corps déformables , 1909, Nature.

[12]  Chunguang Xia,et al.  Broadband acoustic cloak for ultrasound waves. , 2010, Physical review letters.

[13]  A. Norris,et al.  Elastic cloaking theory , 2011, 1103.6045.

[14]  Daniel Torrent,et al.  Acoustic cloaking in two dimensions: a feasible approach , 2008 .

[15]  R. Rivlin,et al.  A uniqueness theorem in the theory of highly-elastic materials , 1948, Mathematical Proceedings of the Cambridge Philosophical Society.

[16]  Leslie Greengard,et al.  Rapid Evaluation of Nonreflecting Boundary Kernels for Time-Domain Wave Propagation , 2000, SIAM J. Numer. Anal..

[17]  L. Treloar,et al.  The elasticity of a network of long-chain molecules.—III , 1943 .

[18]  A. Norris,et al.  Hyperelastic cloaking theory: transformation elasticity with pre-stressed solids , 2012, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[19]  S. Cummer,et al.  One path to acoustic cloaking , 2007 .

[20]  Matti Lassas,et al.  On nonuniqueness for Calderón’s inverse problem , 2003 .

[21]  Non-uniqueness of time-domain reflection from 3D planar elastic layers , 2011 .

[22]  David R. Smith,et al.  Metamaterial Electromagnetic Cloak at Microwave Frequencies , 2006, Science.

[23]  David R. Smith,et al.  Controlling Electromagnetic Fields , 2006, Science.

[24]  Feruza A. Amirkulova,et al.  Active elastodynamic cloaking , 2012, 1211.5558.

[25]  R. S. Rivlin,et al.  Large elastic deformations of isotropic materials. I. Fundamental concepts , 1948, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences.

[26]  R. Rivlin.  Large Elastic Deformations of Isotropic Materials , 1997 .

[27]  Stress-wave energy management through material anisotropy , 2010 .

[28]  U. Leonhardt.  Optical Conformal Mapping , 2006, Science.

[29]  Andrew N. Norris,et al.  Employing pre-stress to generate finite cloaks for antiplane elastic waves , 2012, 1203.3243.

[30]  P. Seppecher,et al.  Transformation Elastodynamics and Active Exterior Acoustic Cloaking , 2011, 1105.1221.

[31]  W. Duan,et al.  Numerical study on a hybrid water wave radiation condition by a 3D boundary element method , 2012 .

[32]  G. Backus.  Long-Wave Elastic Anisotropy Produced by Horizontal Layering , 1962 .

[33]  David Wall,et al.  Partial elastodynamic cloaking by means of fiber-reinforced composites , 2011 .

[34]  S. Guenneau,et al.  Broadband cylindrical acoustic cloak for linear surface waves in a fluid. , 2008, Physical review letters.