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Effective Dynamic Constitutive Relations for 3-D Periodic Elastic Composites

Published online by Cambridge University Press:  10 April 2013

Ankit Srivastava  and
Sia Nemat-Nasser
Ankit Srivastava
Affiliation:
Department of Mechanical and Aerospace Engineering University of California, San Diego La Jolla, CA, 92093-0416 USA
Sia Nemat-Nasser
Affiliation:
Department of Mechanical and Aerospace Engineering University of California, San Diego La Jolla, CA, 92093-0416 USA
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Abstract

Central to the idea of metamaterials is the concept of dynamic homogenization which seeks to define frequency dependent effective properties for Bloch wave propagation. Recent advances in the theory of dynamic homogenization have established the coupled form of the constitutive relation (Willis constitutive relation). This coupled form of the constitutive relation naturally emerges from ensemble averaging of the dynamic fields and automatically satisfies the dispersion relation in the case of periodic composites. Its importance is also notable due to its invariance under transformational acoustics. Here we discuss the explicit form of the effective dynamic constitutive equations. We elaborate upon the existence and emergence of coupling in the dynamic constitutive relation and further symmetries of the effective tensors.

Keywords

compositeacousticelastic properties
Type
Articles
Information
MRS Online Proceedings Library (OPL) , Volume 1508: Symposium BB – Recent Advances in Optical, Acoustic and Other Emerging Metamaterials , 2013 , mrsf12-1508-bb01-12
Copyright
Copyright © Materials Research Society 2013

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References

REFERENCES

Simovski, C. R., “Material parameters of metamaterials (a review), ” Optics and Spectroscopy 107, 726 (2009).10.1134/S0030400X09110101CrossRefGoogle Scholar
Willis, J., “The overall elastic response of composite materials, ” Journal of applied mechanics 50, 1202 (1983).10.1115/1.3167202CrossRefGoogle Scholar
Diener, G., Hurrich, A., and Weissbarth, J., “Bounds on the non-local effective elastic properties of composites, ” Journal of the Mechanics and Physics of Solids 32(1), 21 (1984).10.1016/0022-5096(84)90003-6CrossRefGoogle Scholar
Bakhvalov, N. and Panasenko, G., [Homogenisation: averaging processes in periodic media: mathematical problems in the mechanics of composite materials], Kluwer Academic Publishers (1989).Google Scholar
Willis, J., “Variational and related methods for the overall properties of composites, ” Advances in applied mechanics 21, 1 (1981).10.1016/S0065-2156(08)70330-2CrossRefGoogle Scholar
Willis, J., “Variational principles for dynamic problems for inhomogeneous elastic media, ” Wave Motion 3(1), 1 (1981).10.1016/0165-2125(81)90008-1CrossRefGoogle Scholar
Smith, D. and Pendry, J., “Homogenization of metamaterials by field averaging (invited paper), ” JOSA B 23(3), 391 (2006).10.1364/JOSAB.23.000391CrossRefGoogle Scholar
Amirkhizi, A. and Nemat-Nasser, S., “Microstructurally-based homogenization of electromagnetic properties of periodic media, ” Comptes Rendus Mecanique 336(1-2), 24 (2008).10.1016/j.crme.2007.10.012CrossRefGoogle Scholar
Bensoussan, A., Lions, J., and Papanicolaou, G., [Asymptotic analysis for periodic structures], vol. 5, North Holland (1978).Google Scholar
Sihvola, A., [Electromagnetic mixing formulas and applications], no. 47, Inspec/Iee (1999).10.1049/PBEW047ECrossRefGoogle Scholar
Willis, J., “Exact effective relations for dynamics of a laminated body, ” Mechanics of Materials 41(4), 385 (2009).10.1016/j.mechmat.2009.01.010CrossRefGoogle Scholar
Willis, J., “Effective constitutive relations for waves in composites and metamaterials, ” Proceedings of the Royal Society A: Mathematical, Physical and Engineering Science (2011).10.1098/rspa.2010.0620CrossRefGoogle Scholar
Shuvalov, A., Kutsenko, A., Norris, A., and Poncelet, O., “Effective willis constitutive equations for periodically stratified anisotropic elastic media, ” Proceedings of the Royal Society A: Mathematical, Physical and Engineering Science 467(2130), 1749(2011).10.1098/rspa.2010.0389CrossRefGoogle Scholar
Srivastava, A., and Nemat-Nasser, S., “Overall Dynamic Properties of 3-D Periodic Elastic Composites, ” Proceeding of the Royal Society 468, 269 (2012).10.1098/rspa.2011.0440CrossRefGoogle Scholar