Elastic cloaking theory

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.

[1]  G. Hu,et al.  Design Arbitrary Shaped 2D Acoustic Cloak Without Singularity , 2009, 0907.2282.

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

[3]  J. Lothe,et al.  On the existence of surface‐wave solutions for anisotropic elastic half‐spaces with free surface , 1976 .

[4]  A. Norris,et al.  Wave impedance matrices for cylindrically anisotropic radially inhomogeneous elastic solids , 2010, 1003.5713.

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

[6]  M. C. Pease,et al.  Methods of Matrix Algebra , 1965 .

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

[8]  Huanyang Chen,et al.  Acoustic cloaking and transformation acoustics , 2010 .

[9]  C. Scandrett,et al.  Acoustic cloaking using layered pentamode materials. , 2010, The Journal of the Acoustical Society of America.

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

[11]  William Thomson,et al.  XXI. Elements of a mathematical theory of elasticity , 1856, Philosophical Transactions of the Royal Society of London.

[12]  Huanyang Chen,et al.  Acoustic cloaking in three dimensions using acoustic metamaterials , 2007 .

[13]  Mohamed Farhat,et al.  Cloaking bending waves propagating in thin elastic plates , 2009 .

[14]  A. Norris,et al.  Acoustic metafluids. , 2008, The Journal of the Acoustical Society of America.

[15]  Liang-Wu Cai,et al.  Analysis of Cummer–Schurig acoustic cloaking , 2007 .

[16]  A. Shuvalov.  A sextic formalism for three–dimensional elastodynamics of cylindrically anisotropic radially inhomogeneous materials , 2003, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

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

[18]  P. D. Folkow,et al.  Dynamic higher-order equations for finite rods , 2010 .

[19]  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.

[20]  A. Norris,et al.  Acoustic metafluids made from three acoustic fluids. , 2010, The Journal of the Acoustical Society of America.

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

[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]  Steven A. Cummer,et al.  Material parameters and vector scaling in transformation acoustics , 2008 .

[25]  Fan Yang,et al.  A multilayer structured acoustic cloak with homogeneous isotropic materials , 2008 .

[26]  G. Uhlmann,et al.  Full-Wave Invisibility of Active Devices at All Frequencies , 2006, math/0611185.

[27]  A. Norris.  Acoustic cloaking theory , 2008, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[28]  G. Milton,et al.  Which Elasticity Tensors are Realizable , 1995 .

[29]  David R. Smith,et al.  Scattering theory derivation of a 3D acoustic cloaking shell. , 2008, Physical review letters.

[30]  J. Pendry,et al.  An acoustic metafluid: realizing a broadband acoustic cloak , 2008 .

[31]  G. Milton.  The Theory of Composites , 2002 .