Experimental Evidence of Rotational Elastic Waves in Granular Phononic Crystals

A generalized theory of elasticity, taking into account the rotational degrees of freedom of point bodies constituting a continuum, was proposed at the beginning of the twentieth century by the Cosserat brothers. We report the experimental observation of coupled rotational-translational modes in a noncohesive granular phononic crystal. While absent in the classical theory of elasticity, these elastic wave modes are predicted by the Cosserat theory. However the Cosserat theory fails to predict correctly the dispersion of the elastic modes in granular crystals even in the long-wavelength limit.

Figure 1

(a) Experimental setup. (b) Theoretical dispersion curves for propagation in the $z$ direction of the hcp granular crystal from the discrete lattice model (circles) and their counterparts from the Cosserat continuum model (lines). The cyclic frequency is normalized to the cutoff cyclic frequency ${\omega }^{\prime }=\omega /{\omega }_L^{c,\mathrm{discrete}}$ of the mode $L$.

Figure 2

Comparison of the experimental cutoff frequencies with the theoretical evaluations. (a) Wave transmittance in an aluminum reference block measured with the shear (continuous line) and compressional (dashed line) transducers. The acoustical energy is transmitted in the whole frequency range from low frequencies to 200 kHz. (b) Transmittance measured in the granular crystal under an applied static loading of 20 mN per contact (22 mN at the bottom). The measurement with the shear transducer shows a cutoff frequency close to 150 kHz. The measurement with the compressional transducer shows a cutoff frequency close to 80 kHz. (c) Dependence of the cutoff frequencies measured with the shear (circles) and compressional (squares) transducers on the external loading. Error bars correspond to a 90% confidence interval.

Figure 3

Received signals from a shear transducer (left) and a longitudinal transducer (right) after transmission through a crystal under static loading of 10 mN per contact (12 mN at the bottom). The excited shear pulse is centered on 200 kHz. Normalized temporal signals are shown at the top, the spectrograms at the bottom. Theoretical arrival times of the modes are shown in continuous lines (10 mN per contact) and dashed lines (12 mN per contact).