Simplest Linear Homogeneous Reduced Gyrocontinuum as an Acoustic Metamaterial
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.
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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.
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