Abstract
We consider gyrocontinuum, whose each point-body is an infinitesimal rigid body containing inside an axisymmetric rotor, attached to the body but freely rotating about its axis. Point-bodies of the medium may perform independent translations and rotations of general kind. The proper rotation of their rotors does not cause stresses in the medium. We consider the case of infinitesimal density of inertia tensor both of rotor and carrying body and large proper rotation velocity of the rotor, resulting together in a finite dynamic spin. Rotor inside each point body does not interact with anything but its carrying point body, i.e. its existence only contributes into the kinetic energy but not to the strain energy.We suppose that this medium does not react to the gradient of turn of the carrying bodies, therefore we call it “reduced”. This yields in zero couple stresses. For simplicity we consider the elastic energy of the medium to be isotropic. This is a medium similar to the reduced Cosserat medium but with the kinetic moment consisting of a gyroscopic term. An example of such an artificially made medium could be a medium consisting of interacting light spheres with light but fast rotating rotors inside them. We consider linear motion of the carrying spheres and investigate harmonic waves in this continuum. We see that, similar to isotropic reduced Cosserat medium, longitudinal wave is non-dispersional, and shear-rotational wave has dispersion and one its branch has a band gap. The band gap depend on the dynamic spin of point bodies and can be controlled via it. Note that all the shear harmonic waves in this medium are not plane waves but have polarization, if the direction of propagation is not orthogonal to the rotor axes.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Altenbach H, Zhilin PA (1988) The theory of elastic simple shells (in Russ.). Advances in Mechanics 11(4):107–147
Cosserat E, Cosserat F (1909) Théorie des corps déformables. Hermann, Paris
Eremeyev VA, Pietraszkiewicz W (2012) Material symmetry group of the non-linear polar-elastic continuum. International Journal of Solids and Structures 49(14):1993–2005
Eremeyev VA, Pietraszkiewicz W (2016) Material symmetry group and constitutive equations of micropolar anisotropic elastic solids. Mathematics and Mechanics of Solids 21(2):210–221
Eremeyev VA, Lebedev LP, Altenbach H (2012) Foundations of micropolar mechanics. Springer Science & Business Media, Heidelberg, New York, Dordrecht, London
Eringen AC, Maugin GA (2012) Electrodynamics of Continua, vol I: Foundations and Solid Media. Springer Science & Business Media
Erofeyev VI (2003) Wave Processes in Solids with Microstructure, Series on Stability, Vibration and Control of Systems, Series A, vol 8. World Scientific
Grekova E, Zhilin P (2001) Basic equations of Kelvin’s medium and analogy with ferromagnets. Journal of Elasticity and the Physical Science of Solids 64(1):29–70
Grekova EF (2012) Nonlinear isotropic elastic reduced Cosserat continuum as a possible model for geomedium and geomaterials. Spherical prestressed state in the semilinear material. Journal of Seismology 16(4):695–707
Grekova EF (2016) Plane waves in the linear elastic reduced Cosserat medium with a finite axially symmetric coupling between volumetric and rotational strains. Mathematics and Mechanics of Solids 21(1):73–93
Grekova EF (2018) Waves in elastic reduced Cosserat medium with anisotropy in the term coupling rotational and translational strains or in the dynamic term. In: Dell’Isola F, Eremeyev V, Porubov A (eds) Advances in Mechanics of Microstructured Media and Structures, Springer, Singapore, Advanced Structured Materials, vol 87
Grekova EF, Maugin GA (2005) Modelling of complex elastic crystals by means of multi-spin micromorphic media. International Journal of Engineering Science 43(5):494–519
Grekova EF, Kulesh MA, Herman GC (2009) Waves in linear elastic media with microrotations, Part 2: Isotropic reduced Cosserat model. Bulletin of the Seismological Society of America 99(2B):1423–1428
Ivanova EA (2010) Derivation of theory of thermoviscoelasticity by means of two-component medium. Acta Mechanica 215(1):261–286
Ivanova EA (2014) Description of mechanism of thermal conduction and internal damping by means of two-component Cosserat continuum. Acta Mechanica 225(3):757–795
Ivanova EA (2015) A new model of a micropolar continuum and some electromagnetic analogies. Acta Mechanica 226(3):697–721
Ivanova EA (2017) Description of nonlinear thermal effects by means of a two-component Cosserat continuum. Acta Mechanica 228(6):2299–2346
Kafadar CB, Eringen AC (1971) Micropolar media - I the classical theory. International Journal of Engineering Science 9(3):271–305
Maugin GA (1988) Continuum Mechanics of Electromagnetic Solids, North-Holland Series in Applied Mathematics and Mechanics, vol 33. Elsevier
Palmov V (1964) Fundamental equations of the theory of asymmetric elasticity. Journal of Applied Mathematics and Mechanics 28(3):496–505
Pietraszkiewicz W, Eremeyev VA (2009) On natural strain measures of the non-linear micropolar continuum. International Journal of Solids and Structures 46(3):774–787
Schwartz LM, Johnson DL, Feng S (1984) Vibrational modes in granular materials. Phys Rev Lett 52:831–834
Zhilin PA (2012) Rational Continuum Mechanics (in Russ.). St. Petersburg Polytechnic University
Acknowledgements
The author had a pleasure to meet Prof. Gérard Maugin in 1999. He was interested in her work on Kelvin’s medium where also the analogy between equations of magnetic continua by Prof. Maugin and equations of Kelvin’s medium was established. Even now she remembers vividly this interesting discussion. Later she collaborated on gyrocontinua with Prof. Maugin during her postdoc stay in Laboratoire en Modélisation en Mécanique, Jussieu, Paris. It is an honour for her to dedicate her work on reduced gyrocontinuum to the memory of Prof. Maugin.
This work was supported by the Russian Foundation for Basic Research (grant 17-01-00230), by Spanish Government Agency Ministerio de Economía y Competitividad (project No. FIS2014- 54539-P) and by Andalusian Government (Junta de Andalucía), support for research group FQM- 253.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer International Publishing AG, part of Springer Nature
About this chapter
Cite this chapter
Grekova, E.F. (2018). Simplest Linear Homogeneous Reduced Gyrocontinuum as an Acoustic Metamaterial. In: Altenbach, H., Pouget, J., Rousseau, M., Collet, B., Michelitsch, T. (eds) Generalized Models and Non-classical Approaches in Complex Materials 1. Advanced Structured Materials, vol 89. Springer, Cham. https://doi.org/10.1007/978-3-319-72440-9_19
Download citation
DOI: https://doi.org/10.1007/978-3-319-72440-9_19
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-72439-3
Online ISBN: 978-3-319-72440-9
eBook Packages: EngineeringEngineering (R0)