Hyperelastic cloaking theory: Transformation elasticity with pre-stressed solids

Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences

Volumn 468, Issue 2146, 2012, Pages 2881-2903

  Norris, A N ^a   Parnell, W J ^b  

^a RUTGERS UNIVERSITY   (United States)

^b UNIVERSITY OF MANCHESTER   (United Kingdom)

Author keywords

Cloaking; Elastic waves; Hyperelastic; Pre stress

Indexed keywords

ELASTIC WAVES; FUNCTIONS; STRAIN ENERGY;

ANTI-PLANE; CLOAKING; CONSTITUTIVE PARAMETERS; DEFORMATION GRADIENTS; HYPER ELASTIC; INCREMENTAL MOTION; ISOTROPIC ELASTICITY; NUMERICAL EXAMPLE; PRE-STRAIN; PRE-STRESS; PRE-STRESSED; SEMILINEAR; STRAIN ENERGY FUNCTIONS; SYMMETRY CONSTRAINTS;

ELASTICITY;

EID: 84866397939     PISSN: 13645021     EISSN: 14712946     Source Type: Journal    
DOI: 10.1098/rspa.2012.0123     Document Type: Article

References (35)

1. Beatty, M. F. & Hayes, M. A. 1992 Deformations of an elastic, internally constrained material. Part 1: homogeneous deformations. J. Elast. 29, 1-84. (doi:10.1007/BF00043445)
2. Bell, J. F. 1985 Contemporary perspectives in finite strain plasticity. Int. J. Plast. 1, 3-27. (doi:10.1016/0749-6419(85)90011-7)
3. Bell, J. F. 1989 Experiments on the kinematics of large plastic strain in ordered solids. Int. J. Solids Struct. 25, 267-277. (doi:10.1016/0020-7683(89) 90048-6)
4. Brun, M., Guenneau, S. & Movchan, A. B. 2009 Achieving control of in-plane elastic waves. Appl. Phys. Lett. 94, 061903. (doi:10.1063/1.3068491)
5. Chen, H. & Chan, C. T. 2007 Acoustic cloaking in three dimensions using acoustic metamaterials. Appl. Phys. Lett. 91, 183518. (doi:10.1063/1. 2803315)
6. Cosserat, E. & Cosserat, F. 1909 Théorie des Corps Déformables. Paris, France: Librairie Scientifique A. Hermann et fils. (ISBN 978-1429704847, Reprinted by Cornell University Library).
7. Cummer, S. A. & Schurig, D. 2007 One path to acoustic cloaking. New J. Phys. 9, 45. (doi:10.1088/ 1367-2630/9/3/045)
8. Cummer, S. A., Popa, B. I., Schurig, D., Smith, D. R., Pendry, J., Rahm, M. & Starr, A. 2008 Scattering theory derivation of a 3D acoustic cloaking shell. Phys. Rev. Lett. 100, 024301. (doi:10.1103/PhysRevLett.100.024301)
9. Dill, E. H. 2007 Continuum mechanics: elasticity, plasticity, viscoelasticity. Boca Raton, FL: CRC Press.
10. Eringen, A. C. & Suhubi, E. S. 1975 Elastodynamics, vol. II, New York, NY: Academic Press.
11. Greenleaf, A., Lassas, M. & Uhlmann, G. 2003 On nonuniqueness for Calderon's inverse problem. Math. Res. Lett. 10, 685-693.
12. Greenleaf, A., Kurylev, Y., Lassas, M. & Uhlmann, G. 2007 Full-wave invisibility of active devices at all frequencies. Commun. Math. Phys. 275, 749-789. (doi:10.1007/s00220-007-0311-6)
13. Jafari, A. H., Abeyaratne, R. & Horgan, C. O. 1984 The finite deformation of a pressurized circular tube for a class of compressible materials. Z. Angew. Math. Phys. 35, 227-246. (doi:10.1007/BF0 0947935)
14. John, F. 1960 Plane strain problems for a perfectly elastic material of harmonic type. Commun. Pure Appl. Math. 13, 239-296. (doi:10.1002/cpa. 3160130206)
15. Lewis, T. S., Kraft, D. W. & Hom, N. 1976 Scattering of elastic waves by a cylindrical cavity in a solid. J. Appl. Phys. 47, 1795-1798. (doi:10.1063/1.322894)
16. Lur'e, A. I. 1968 Theory of elasticity for a semilinear material. Prikl. Mat. Mekh (J. Appl. Math. Mech.) 32, 1068-1085. (doi:10.1016/0021-8928(68)90034- 8)
17. Martin, P. A. 2006 Multiple scattering: interaction of time-harmonic waves with N obstacles. New York, NY: Cambridge University Press. (doi:10.1017/CBO9780511735110)
18. Milton, G. W. & Willis, J. R. 2007 On modifications of Newton's second law and linear continuum elastodynamics. Proc. R. Soc. A 463, 855-880. (doi:10.1098/rspa.2006.1795)
19. Milton, G. W., Briane, M. & Willis, J. R. 2006 On cloaking for elasticity and physical equations with a transformation invariant form. New J. Phys. 8, 248-267. (doi:10.1088/1367-2630/8/10/248)
20. Norris, A. N. 2008 Acoustic cloaking theory. Proc. R. Soc. A 464, 2411-2434. (doi:10.1098/rspa.20 08.0076)
21. Norris, A. N. 2009 Acoustic metafluids. J. Acoust. Soc. Am. 125, 839-849. (doi:10.1121/1.3050288)
22. Norris, A. N. & Shuvalov, A. L. 2011 Elastic cloaking theory. Wave Motion 49, 525-538. (doi:10.1016/j.wavemoti.2011.03.002)
23. Ogden, R. W. 1997 Non-linear elastic deformations. New York, NY: Dover Publications.
24. Ogden, R. W. 2007 Incremental statics and dynamics of pre-stressed elastic materials. In Waves in nonlinear pre-stressed materials: CISM Int. Centre for Mechanical Sciences (eds M. Destrade & G. Saccomandi), vol. 495, ch. 1, pp. 1-26. Berlin, Germany: Springer.
25. Olsson, P. 2011 Non-uniqueness of time-domain reflection from 3D planar elastic layers. Wave Motion 48, 539-549. (doi:10.1016/j.wavemoti.2011.04.009)
26. Olsson, P. & Wall, D. J. N. 2011 Partial elastodynamic cloaking by means of fiber-reinforced composites. Inverse Prob. 27, 045010. (doi:10.1088/0266-5611/27/4/045010)
27. Parnell, W. J. 2012 Nonlinear pre-stress for cloaking from antiplane elastic waves. Proc. R. Soc. A 468, 563-580. (doi:10.1098/rspa.2011.0477)
28. Parnell, W. J., Norris, A. N. & Shearer, T. 2012 Employing pre-stress to generate finite cloaks for antiplane elastic waves. Appl. Phys. Lett. 100, 171907. (doi:10.1063/1.4704566)
29. Pendry, J. B., Schurig, D. & Smith, D. R. 2006 Controlling electromagnetic fields. Science 312, 1780-1782. (doi:10.1126/science.1125907)
30. Raasch, I. 1975 A variational principle for non-linear elastodynamics and its application to the hybrid stress model. Int. J. Nonlinear Mech. 10, 215-222. (doi:10.1016/0020-7462(75)90013-X)
31. Schurig, D., Mock, J. J., Justice, B. J., Cummer, S. A., Pendry, J. B., Starr, A. F. & Smith, D. R. 2006 Metamaterial electromagnetic cloak at microwave frequencies. Science 314, 977-980. (doi:10.1126/science.1133628)
32. Sensenig, C. B. 1964 Instability of thick elastic solids. Commun. Pure Appl. Math. 17, 451-491. (doi:10.1002/cpa.3160170406)
33. Vasquez, F. G., Milton, G. W., Onofrei, D. & Seppecher, P. 2012 Transformation elastodynamics and active exterior acoustic cloaking. In Acoustic metamaterials: negative refraction, imaging, lensing and cloaking (eds. S. Guenneau and R. Craster), pp. 1-1. Bristol, UK: Canopus Academic Publishing and Springer SBM.
34. Willis, J. R. 1997 Dynamics of composites. In Continuum Micromechanics: CISM Lecture Notes, no. 495, ch. 1, pp. 265-290. New York, NY: Springer.
35. Zubov, L. M. 1970 The stationary principle of complementary work in non-linear theory of elasticity. Prikl. Mat. Mekh (J. Appl. Math. Mech.) 34, 241-245. (doi:10.1016/0021-8928(70)90 136-X)