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Register a new userCalculations of the Structure of Basin Volumes for Mechanically Stable Packings
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There are a finite number of distinct mechanically stable (MS) packings in model granular systems composed of frictionless spherical grains. For typical packing-generation protocols employed in experimental and numerical studies, the probabilities with which the MS packings occur are highly nonuniform and depend strongly on parameters in the protocol. Despite intense work, it is extremely difficult to predict {it a priori} the MS packing probabilities, or even which MS packings will be the most versus the least probable. We describe a novel computational method for calculating the MS packing probabilities by directly measuring the volume of the MS packing `basin of attraction, which we define as the collection of initial points in configuration space at {it zero packing fraction} that map to a given MS packing by following a particular dynamics in the density landscape. We show that there is a small core region with volume $V^c_n$ surrounding each MS packing $n$ in configuration space in which all initial conditions map to a given MS packing. However, we find that the MS packing probabilities are very weakly correlated with core volumes. Instead, MS packing probabilities obtained using initially dilute configurations are determined by complex geometric features of the basin of attraction that are distant from the MS packing.
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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.
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.
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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