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Register a new userFinite Element Simulation of Dense Wire Packings
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A finite element program is presented to simulate the process of packing and coiling elastic wires in two- and three-dimensional confining cavities. The wire is represented by third order beam elements and embedded into a corotational formulation to capture the geometric nonlinearity resulting from large rotations and deformations. The hyperbolic equations of motion are integrated in time using two different integration methods from the Newmark family: an implicit iterative Newton-Raphson line search solver, and an explicit predictor-corrector scheme, both with adaptive time stepping. These two approaches reveal fundamentally different suitability for the problem of strongly self-interacting bodies found in densely packed cavities. Generalizing the spherical confinement symmetry investigated in recent studies, the packing of a wire in hard ellipsoidal cavities is simulated in the frictionless elastic limit. Evidence is given that packings in oblate spheroids and scalene ellipsoids are energetically preferred to spheres.
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We describe a series of experiments involving the creation of cylindrical packings of star-shaped particles, and an exploration of the stability of these packings. The stars cover a broad range of arm sizes and frictional properties. We carried out three different kinds of experiments, all of which involve columns that are prepared by raining star particles one-by-one into hollow cylinders. As an additional part of the protocol, we sometimes vibrated the column before removing the confining cylinder. We rate stability in terms of r, the ratio of the mass of particles that fall off a pile when it collapsed, to the total particle mass. The first experiment involved the intrinsic stability of the pile when the confining cylinder was removed. The second kind of experiment involved adding a uniform load to the top of the column, and then determining the collapse properties. A third experiment involved testing stability to tipping of the piles. We find a stability diagram relating the pile height, h, vs. pile diameter, delta, where the stable and unstable regimes are separated by a boundary that is roughly a power-law in h vs. delta with an exponent that is less than one. Increasing friction and vibration both tend to stabilize piles, while increasing particle size can destabilize the system under certain conditions.
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 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.
For packings of hard but not perfectly rigid particles, the length scales that govern the packing geometry and the contact forces are well separated. This separation of length scales is explored in the force network ensemble, where one studies the space of allowed force configurations for a given, frozen contact geometry. Here we review results of this approach, which yields nontrivial predictions for the effect of packing dimension and anisotropy on the contact force distribution $P(f)$, the response to overall shear and point forcing, all of which can be studied in great numerical detail. Moreover, there are emerging analytical approaches that very effectively capture, for example, the form of force distributions.
Using X-ray tomography, we experimentally investigate granular packings subject to mechanical tapping for three types of beads with different friction coefficients. We validate Edwards volume ensemble in these three-dimensional granular systems and establish a granular version of thermodynamic zeroth law. Within Edwards framework, we also explicitly clarify how friction influences granular statistical mechanics as modifying the density of states, which allows us to determine the entropy as a function of packing fraction and friction subsequently. Additionally, we obtain a granular jamming phase diagram based on geometric coordination number and packing fraction.
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