Linear and Nonlinear Waves in Magneto-Granular Phononic Structures (PhD Thesis)

Ph.D Defense on June 14th 2017

Resume

Granular crystals are periodic or structured arrangements of elastic particles in contact. This work is devoted to theoretical and experimental study of the elastic wave propagation through such structures. A one-dimensional granular crystal composed of steel spherical beads coupled to permanent magnets placed in a substrate is first studied. Static forces at the contact between beads, determining the wave propagation and dispersion characteristics, are induced by the magnetic field from the magnets. This configuration enables tuning the dynamic response of the chain by modifying the magnetic strength of the magnets. A linear model taking into account all degrees of freedom of the beads (three translations and three rotations) as well as all elastic couplings (longitudinal, shear and torsional), between the beads and between the beads and the substrate is developed. This model provides the dispersion relations of the modes in the system for different coupling parameters. The associated experiments confirm the elastic propagation of modes with micro-rotation of beads and demonstrate the pertinence of the model for the system description. Several interesting effects on the dispersion are observed and discussed, zero group velocity modes, soft modes. . .

In a second part, we take into account the nonlinearities originating from the contacts to predict and then observe experimentally the second harmonic generation. The filtering of harmonics along with conversion from longitudinal to coupled transversal-rotational modes in granular structures, is also observed for a configuration deviating from simple one-dimensional chains. This work opens the way for interesting applications in elastic wave control, in the nonlinear regime, with structured granular devices.

Keywords: granular phononic crystal, rotational modes, nonlinearity, granular structures, wave control.

PhD Supervisors:

Director : Vincent Tournat, CNRS Senior Research Scientist LAUM, University of Le Mans

Co-Director : Vitalyi Gusev, Professor LAUM, University of Le Mans

Scientific Supervisor : Georgios Theocharis, CNRS Research Scientist LAUM, University of Le Mans

Articles related to this work:

1 - F. Allein, V. Tournat, V. E. Gusev, and G. Theocharis,

Tunable magneto-granular phononic crystals,

Applied Physics Letters 108, 161903 (2016). PDF

DOI: https://doi.org/10.1063/1.4947192

2 - F. Allein, V. Tournat, V. E. Gusev, and G. Theocharis,

Transversal-rotational and zero group velocity modes in tunable magneto-granular phononic crystals,

Extreme Mechanics Letters, 12:65-70 (2017). PDF

DOI: https://doi.org/10.1016/j.eml.2016.08.001

3 - F. Allein, V. Tournat, V. Gusev, and G. Theocharis,

Linear and Nonlinear Elastic Waves in Magnetogranular Chains,

Physical Review Applied 13, 024023 (2020). PDF

DOI: https://doi.org/10.1103/PhysRevApplied.13.024023

Paper selected as an "Editors' Suggestion"

Ph.D. manuscript: PDF

(a) Schematic diagram of the granular chain on top of permanent magnets within a rubber. Forces and magnetic flux lines induced by the magnetic field have been superimposed. (b) Magnetic flux density (mT) vs. distance from the upper surface of the magnets. (c) Schematic diagram of the experimental setup.

Dispersion curves (left) and experimental linear spectrum for a chain of 5 beads (center) and 15 beads (right).

Representation of the infinite free granular chain by mass-spring lattice.