• Merkel, Aurélien
• Tournat, Vincent
• Goussev, Vitali

Abstract

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. One hundred years ago, the Cosserat brothers developed a continuum elasticity theory accounting for the rotational degrees of freedom of point bodies (infinitesimal particles) constituting deformable solids [1]. Currently, this theory is known as the Cosserat theory, and the related and advanced theories are known as theories of Cosserat continuum or as theories of micropolar continuum [2]. In the Cosserat theory, each material point has 6 degrees of freedom, three of which correspond to the translations as in the classical theory of elasticity, and the three others correspond to rotations. The stress tensor is asymmetric and an additional couple-stress tensor is introduced, which plays the same role for torques as the stress tensor plays for forces. The theory predicts a contribution of rotations to the dispersion of the shear elastic wave velocity as well as the existence of additional rotational wave modes [3]. For the description of an isotropic medium, the classical elasticity requires the knowledge of the elastic constants and (the Lamé constants) and the material density. The Cosserat theory requires four additional elastic constants , ", , , describing the microstructure, and the density of the moment of inertia J [2,4]. However, even 100 years after the Cosserats, the experimental evidence of the Cosserat effects are rare, difficult to obtain, and subject to criticism [2,3,5]. The additional rota-tional mode resonances expected in the elastic vibration of macroscopic micropolar bodies [6] have never been observed. The determination and calibration of the additional Cosserat elastic constants from static or quasisatic experiments is delicate. In our opinion, the most crucial observation providing convincing evidence of the effects of the rotational degrees of freedom, the milestone of the Cosserat theory, would be the observation of the propagation of rotational waves in elastically micro-inhomogeneous media. The only cited observation [2,3] of rotational waves in a specimen composed of metallic particles randomly distributed in an epoxy polymer matrix [7] is not convincing [8]. The rotational waves in the Cosserat continuum theory is often considered a mathematical possibility, but proving their actual existence by experiments constitutes a serious endeavor [9]. Below, we demonstrate that important advantages for testing the fundamentals of the Cosserat theory can be gained through conducting the experiments in noncohesive phononic granular crystals which are not saturated by liquids. In noncohesive granular media, the macroscopic dynamical behavior depends on the peculiar shear and normal contact interactions between the spheres at the microscopic level, which are well described by the Hertz-Mindlin theory of contact [10,11]. According to this theory , in the first aproximation, a weakly precompressed granular crystal, where the diameters d of the contacts between the beads are much smaller than the diameter a of the beads (d=a (1), can be reduced to a mass-spring structure. Since the size of the grains is comparable to the distance between neighbors, the rotational degrees of freedom of the beads must be taken into account in order to accurately describe the dynamics of the granular media [12,13]. However, while the bending rigidity of the contacts [14] and their spin (torsional) rigidity generally contribute to the effects of the rotation of the beads [9,15], their contribution is not needed here. The moment [1,3] applied to the beads due to these two rigidities is about ða=dÞ 2) 1 times smaller than the moment applied to the beads due to the shear rigidity. In the Cosserat continuum, the absence of torsional and bending rigidities corresponds to negligible nonclassical elastic constants ", , and and to the absence of the momentum stresses acting on the contacts between the beads. Thus, the only nonclassical elastic constant expected to play an important role in non-cohesive granular crystals is , which introduces asymmetry of the stress tensor. These circumstances provide important simplifications in identifying the contribution of rotational degrees of freedom to elastic wave motion PRL

Topics

• density
• impedance spectroscopy
• dispersion
• polymer
• grain
• theory
• experiment
• elasticity
• isotropic