Analytical formulation of three-dimensional dynamic homogenization for periodic elastic systems

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

Volumn 468, Issue 2142, 2012, Pages 1629-1651

  Norris, A N ^a   Shuvalov, A L ^b   Kutsenko, A A ^b  

^a RUTGERS UNIVERSITY   (United States)

^b UNIVERSITÉ DE BORDEAUX   (France)

Author keywords

Bloch spectrum; Homogenization; Periodic elastic medium

Indexed keywords

EQUATIONS OF MOTION; THREE DIMENSIONAL;

ANALYTICAL FORMULATION; ARBITRARY FREQUENCIES; BLOCH SPECTRUM; BLOCH WAVES; EFFECTIVE EQUATIONS; EFFECTIVE MEDIUM THEORIES; EFFECTIVE MODULI; ELASTIC MEDIUM; ELASTIC SYSTEMS; ELASTO-DYNAMICS; FREQUENCY DEPENDENCE; HOMOGENIZATION; MATERIAL PARAMETER; NUMERICAL EXAMPLE; ONE-DIMENSIONAL SYSTEMS; PERIODIC MEDIUM; PLANE WAVE EXPANSION METHOD; THREE-DIMENSIONAL DYNAMICS; WAVE NUMBERS;

ELASTICITY;

EID: 84862188899     PISSN: 13645021     EISSN: 14712946     Source Type: Journal    
DOI: 10.1098/rspa.2011.0698     Document Type: Conference Paper

References (20)

1. Avila, A., Griso, G. & Miara, B. 2005 Bandes phononiques interdites en élasticité linéarisée. C. R. Acad. Sci. Paris, Ser. I 340, 933-938. (Pubitemid 40912042)
2. Avila, A., Griso, G., Miara, B. & Rohan, E. 2008 Multiscale modeling of elastic waves: theoretical justification and numerical. Multiscale Model. Simul. 7, 1-21. (doi:10.1137/060677689)
3. Liu, Z., Chan, C. T. & Sheng, P. 2005 Analytic model of phononic crystals with local resonances. Phys. Rev. B 71, 014103. (doi:10.1103/PhysRevB. 71.014103) (Pubitemid 40486264)
4. 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)
5. 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) (Pubitemid 44631329)
6. Nemat-Nasser, S. & Srivastava, A. 2011 Overall dynamic constitutive relations of layered elastic composites. J. Mech. Phys. Solids 59, 1953-1956. (doi:10.1016/j.jmps.2011.07.008)
7. Ruppin, R. 2002 Electromagnetic energy density in a dispersive and absorptive material. Phys. Lett. A 299, 309-312. (doi:10.1016/S0375-9601(01) 00838-6)
8. Schoenberg, M. & Sen, P. N. 1983 Properties of a periodically stratified acoustic half-space and its relation to a Biot fluid. J. Acoust. Soc. Am. 73, 61-67. (doi:10.1121/1.388724) (Pubitemid 13507244)
9. Shuvalov, A. L., Kutsenko, A. A. & Norris, A. N. 2010 Divergence of the logarithm of a unimodular monodromy matrix near the edges of the Brillouin zone. Wave Motion 47, 370-382. (doi:10.1016/j.wavemoti.2009.12.005)
10. Shuvalov, A. L., Kutsenko, A. A., Norris, A. N. & Poncelet, O. 2011 Effective Willis constitutive equations for periodically stratified anisotropic elastic media. Proc. R. Soc. A 467, 1749-1769. (doi:10.1098/rspa.2010.0389)
11. Srivastava, A. & Nemat-Nasser, S. 2012 Overall dynamic properties of three-dimensional periodic elastic composites. Proc. R. Soc. A 468, 269-287. (doi:10.1098/rspa.2011.0440)
12. Willis, J. R. 1980 Polarization approach to the scattering of elastic waves-I. Scattering by a single inclusion. J. Mech. Phys. Solids 28, 287-305. (doi:10.1016/0022-5096(80)90021-6) (Pubitemid 11478846)
13. Willis, J. R. 1981a Variational and related methods for the overall properties of composites. Adv. Appl. Mech. 21, 1-78. (doi:10.1016/S0065-2156(08) 70330-2) (Pubitemid 12465507)
14. Willis, J. R. 1981b Variational principles and operator equations for electromagnetic waves in inhomogeneous media. Wave Motion 6, 127-139. (doi:10.1016/0165-2125(84)90009-X)
15. Willis, J. R. 1983 The overall elastic response of composite materials. J. Appl. Mech. ASME 50, 1202-1209. (doi:10.1115/1.3167202)
16. Willis, J. R. 1984 Variational principles for dynamic problems for inhomogeneous elastic media. Wave Motion 3, 1-11. (doi:10.1016/0165-2125(81) 90008-1)
17. Willis, J. R. 1985 The non-local influence of density variations in a composite. Int. J. Solids Struct. 210, 805-817. (doi:10.1016/0020-7683(85)90084- 8) (Pubitemid 15562799)
18. Willis, J. R. 1997 Dynamics of composites, vol. 495. Continuum Micromechanics: CISM Lecture Notes, ch. 1, pp. 265-290. Vienna, Austria: Springer.
19. Willis, J. R. 2009 Exact effective relations for dynamics of a laminated body. Mech. Materials 41, 385-393. (doi:10.1016/j.mechmat.2009.01.010)
20. Willis, J. R. 2011 Effective constitutive relations for waves in composites and metamaterials. Proc. R. Soc. A 467, 1865-1879. (doi:10.1098/rspa.2010.0620)