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Register a new userConstraints and vibrations in static packings of ellipsoidal particles
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We numerically investigate the mechanical properties of static packings of ellipsoidal particles in 2D and 3D over a range of aspect ratio and compression $Delta phi$. While amorphous packings of spherical particles at jamming onset ($Delta phi=0$) are isostatic and possess the minimum contact number $z_{rm iso}$ required for them to be collectively jammed, amorphous packings of ellipsoidal particles generally possess fewer contacts than expected for collective jamming ($z < z_{rm iso}$) from naive counting arguments, which assume that all contacts give rise to linearly independent constraints on interparticle separations. To understand this behavior, we decompose the dynamical matrix $M=H-S$ for static packings of ellipsoidal particles into two important components: the stiffness $H$ and stress $S$ matrices. We find that the stiffness matrix possesses $N(z_{rm iso} - z)$ eigenmodes ${hat e}_0$ with zero eigenvalues even at finite compression, where $N$ is the number of particles. In addition, these modes ${hat e}_0$ are nearly eigenvectors of the dynamical matrix with eigenvalues that scale as $Delta phi$, and thus finite compression stabilizes packings of ellipsoidal particles. At jamming onset, the harmonic response of static packings of ellipsoidal particles vanishes, and the total potential energy scales as $delta^4$ for perturbations by amplitude $delta$ along these `quartic modes, ${hat e}_0$. These findings illustrate the significant differences between static packings of spherical and ellipsoidal particles.
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We investigate the mechanical response of jammed packings of circulo-lines, interacting via purely repulsive, linear spring forces, as a function of pressure $P$ during athermal, quasistatic isotropic compression. Prior work has shown that the ensemble-averaged shear modulus for jammed disk packings scales as a power-law, $langle G(P) rangle sim P^{beta}$, with $beta sim 0.5$, over a wide range of pressure. For packings of circulo-lines, we also find robust power-law scaling of $langle G(P)rangle$ over the same range of pressure for aspect ratios ${cal R} gtrsim 1.2$. However, the power-law scaling exponent $beta sim 0.8$-$0.9$ is much larger than that for jammed disk packings. To understand the origin of this behavior, we decompose $langle Grangle$ into separate contributions from geometrical families, $G_f$, and from changes in the interparticle contact network, $G_r$, such that $langle G rangle = langle G_frangle + langle G_r rangle$. We show that the shear modulus for low-pressure geometrical families for jammed packings of circulo-lines can both increase {it and} decrease with pressure, whereas the shear modulus for low-pressure geometrical families for jammed disk packings only decreases with pressure. For this reason, the geometrical family contribution $langle G_f rangle$ is much larger for jammed packings of circulo-lines than for jammed disk packings at finite pressure, causing the increase in the power-law scaling exponent.
We present experimental and numerical results for displacement response functions in packings of rigid frictional disks under gravity. The central disk on the bottom layer is shifted upwards by a small amount, and the motions of disks above it define the displacement response. Disk motions are measured with the help of a still digital camera. The responses so measured provide information on the force-force response, that is, the excess force at the bottom produced by a small overload in the bulk. We find that, in experiments, the vertical-force response shows a Gaussian-like shape, broadening roughly as the square root of distance, as predicted by diffusive theories for stress propagation in granulates. However, the diffusion coefficient obtained from a fit of the response width is ten times larger than predicted by such theories. Moreover we notice that our data is compatible with a crossover to linear broadening at large scales. In numerical simulations on similar systems (but without friction), on the other hand, a double-peaked response is found, indicating wave-like propagation of stresses. We discuss the main reasons for the different behaviors of experimental and model systems, and compare our findings with previous works.
We report numerical results of effective attractive forces on the packing properties of two-dimensional elongated grains. In deposits of non-cohesive rods in 2D, the topology of the packing is mainly dominated by the formation of ordered structures of aligned rods. Elongated particles tend to align horizontally and the stress is mainly transmitted from top to bottom, revealing an asymmetric distribution of local stress. However, for deposits of cohesive particles, the preferred horizontal orientation disappears. Very elongated particles with strong attractive forces form extremely loose structures, characterized by an orientation distribution, which tends to a uniform behavior when increasing the Bond number. As a result of these changes, the pressure distribution in the deposits changes qualitatively. The isotropic part of the local stress is notably enhanced with respect to the deviatoric part, which is related to the gravity direction. Consequently, the lateral stress transmission is dominated by the enhanced disorder and leads to a faster pressure saturation with depth.
Contact breaking and Hertzian interactions between grains can both give rise to nonlinear vibrational response of static granular packings. We perform molecular dynamics simulations at constant energy in 2D of frictionless bidisperse disks that interact via Hertzian spring potentials as a function of energy and measure directly the vibrational response from the Fourier transform of the velocity autocorrelation function. We compare the measured vibrational response of static packings near jamming onset to that obtained from the eigenvalues of the dynamical matrix to determine the temperature above which the linear response breaks down. We compare packings that interact via single-sided (purely repulsive) and double-sided Hertzian spring interactions to disentangle the effects of the shape of the potential from contact breaking. Our studies show that while Hertzian interactions lead to weak nonlinearities in the vibrational behavior (e.g. the generation of harmonics of the eigenfrequencies of the dynamical matrix), the vibrational response of static packings with Hertzian contact interactions is dominated by contact breaking as found for systems with repulsive linear spring interactions.
The effects of particle shape on the vibrational properties of colloidal glasses are studied experimentally. Ellipsoidal glasses are created by stretching polystyrene spheres to different aspect ratios and then suspending the resulting ellipsoidal particles in water at high packing fraction. By measuring displacement correlations between particles, we extract vibrational properties of the corresponding shadow ellipsoidal glass with the same geometric configuration and interactions as the source suspension but without damping. Low frequency modes in glasses composed of ellipsoidal particles with major/minor axis aspect ratios $sim$1.1 are observed to have predominantly rotational character. By contrast, low frequency modes in glasses of ellipsoidal particles with larger aspect ratios ($sim$3.0) exhibit a mix of rotational and translational character. All glass samples were characterized by a distribution of particles with different aspect ratios. Interestingly, even within the same sample it was found that small-aspect-ratio particles participate relatively more in rotational modes, while large-aspect-ratio particles tend to participate relatively more in translational modes.
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