^{1}, Aroune Duclos

^{1}, Jean-Philippe Groby

^{1}, Vincent Tournat

^{2}and Vitalyi Gusev

^{3}

^{1}Laboratory of Acoustics at University of Maine.

^{2}Lunam Université, Université du Maine, CNRS

^{3}Université du Maine.

The question of the existence/inexistence of localized vibrations and surface acoustic waves is of large interest in mechanics and acoustics because of the important role played by these modes in various physical processes. It is well known that in a semi-infinite linear chain of masses coupled by harmonic compression/dilatation springs, the vibrations cannot be localized near the edge if all the masses and all the springs are equal. However, in this one-dimensional (1-D) granular crystal with a single degree of freedom, i.e., with only displacements along the chain axis being allowed, the localized vibrations exist when the masses take alternatively two different values and the chain starts from a lighter mass [1].

We demonstrate theoretically that localized edge mode exists in a 1-D granular phononic crystal composed of infinitely long cylinders, even if they have equal masses, when the contacts between the cylinders provide linear shear/transversal and linear bending rigidities. The considered mechanical system has two degrees of freedom, because the masses/cylinders can move normally to the axes of the cylinders and of the chain, but also can rotate. Two dispersive propagating acoustic modes, in which the rotation and translation motions are mixed, are predicted in this granular phononic crystal. These modes are separated by a gap of forbidden frequencies. The theory predicts that, if the localization occurs, the frequency of the localized vibration, composed of the two evanescent acoustic modes, is located inside the propagating frequencies gap.

By considering a semi-infinite chain with a free boundary condition, we establish the necessary conditions for the existence of a localized mode. Simple analytical expressions are obtained for the propagating and localized modes, which provide opportunity for a straightforward evaluation of the existence and the frequency of the localized mode depending on the relative strength of the shear and bending inter-particle interactions. When the structure is not composed of empty cylindrical shells, and if a band gap exists between the two propagating bands, then localization is found for any values of shear and bending rigidities.

We show the influence of the shear and bending rigidities of the contacts on the localization depth. The amplitude of the transversal and rotational displacements is the largest at the edge, i.e, at the first particle, and decreases exponentially along the chain.

These results are complementary to recent theoretical [2,3] and experimental [4] investigations of the acoustic waves in 2-D and 3-D granular crystals possessing rotational degrees of freedom. These investigations are conducted in view of their potential application as phononic metamaterials for shear wave control.