Orientational ordering in crumpled elastic sheets

We report an experimental study of the development of orientational order in a crumpled sheet, with a particular focus on the role played by the geometry of confinement. Our experiments are performed on elastomeric sheets immersed in a fluid, so that the effects of plasticity and friction are suppressed. When the sheet is crumpled either axially or radially within a cylinder, we find that the sheet aligns with the flat or the curved wall, depending on the aspect ratio of the cylinder. Nematic correlations develop between the normals of the sheets at relatively low volume fractions and the crumpled object has large density fluctuations corresponding to the stacking of parallel sheets. The aligning effect of the wall breaks symmetry and selects the direction of ordering

Kepler orbits of settling discs

The collective dynamics of objects moving through a viscous fluid is complex and counterintuitive. A key to understanding the role of nontrivial particle shape in this complexity is the interaction of a pair of sedimenting spheroids. We report experimental results on two discs settling at negligible Reynolds number ($\simeq 10^{-4}$), finding two classes of bound periodic orbits, each with transitions to scattering states. We account for these dynamics, at leading far-field order, through an effective Hamiltonian in which gravitational driving endows orientation with the properties of momentum. This leads to a precise correspondence with the Kepler problem of planetary motion for a wide range of initial conditions, and also to orbits with no Keplerian analogue. This notion of internal degrees of freedom manifesting themselves as an effective inertia is potentially a more general tool in Stokesian driven systems

3-dimensional structure of a sheet crumpled into a ball

When a thin sheet is crushed into a small three-dimensional volume, it invariably forms a structure with a low volume fraction but high resistance to further compression. Being a far-from-equilibrium process, forced crumpling is not necessarily amenable to a statistical description in which the parameters of the initially flat sheet and the final confinement fully specify the resulting crumpled state. Instead, the internal geometry and mechanical properties of the crumpled ball may reflect the history of its preparation. Our X-ray microtomography experiments reveal that the internal 3-dimensional geometry of a crumpled ball is in many respects isotropic and homogeneous. In these respects, crumpling recapitulates other classic nonequilibrium problems such as turbulence, where a system driven by long-wavelength, low-symmetry, forcing shows only rather subtle fingerprints of the forcing mechanism. However, we find local nematic ordering of the sheet into parallel stacks. The layering proceeds radially inwards from the outer surface. The extent of this layering increases with the volume fraction, or degree of compression.Comment: Accepted by the PNAS Editorial Board July 16, 2011 (received for review December 20, 2010

Large Force Fluctuations in a Flowing Granular Medium

We report the characteristics of the temporal fluctuations in the local force delivered to the wall of a 2D hopper by a granular medium flowing through it. The forces are predominantly impulsive at all flow rates for which the flow does not permanently jam. The average impulse delivered to the wall is much larger than the momentum acquired by a single particle under gravity between collisions, reflecting the fact that momentum is transferred to the walls from the bulk of the flow by collisions. At values larger than the average impulse, the probability distribution of impulses is broad and decays exponentially on the scale of the average impulse, just as it does in static granular media. At small impulse values, the probability distribution evolves continuously with flow velocity but does not show a clear signature of the transition from purely collisional flow to intermittently jamming flows. However, the time interval between collisions tends to a power law distribution, $P(\tau)\sim \tau^{-3/2}$, thus showing a clear dynamical signature of the approach to jamming.Comment: 4 pages, 3 figure

Force-velocity correlations in a dense, collisional, granular flow

We report measurements in a 2-dimensional, gravity-driven, collisional, granular flow of the normal force delivered to the wall and of particle velocity at several points in the flow. The wall force and the flow velocity are negatively correlated. This correlation falls off only slowly with distance transverse to the flow, but dies away on the scale of a few particle diameters upstream or downstream. The data support a picture of short-lived chains of frequently colliding particles that extend transverse to the flow direction, making transient load-bearing bridges that cause bulk fluctuations in the flow velocity. The time-dependence of these spatial correlation functions indicate that while the force-bearing structures are local in space, their influence extends far upstream in the flow, albeit with a time-lag. This leads to correlated velocity fluctuations, whose spatial range increases as the jamming threshold is approached.Comment: to be submitted for publicatio

Learning versus Diversification in Project Choice

We study the issue of project choice when a risk-averse agent must choose whether to invest in two projects of the same type (focus) or of different types (diversification). Projects of the same type are subject to common type-specific shocks. Hence focusing is more risky within each period, but enables faster learning across periods. Optimal project choice involves balancing these two considerations. We demonstrate how an agent's choice of whether to focus or diversify is related to (i) the speed of learning (ii) the type-specific risk and (iii) his risk- aversion and investment horizon. We show that, contrary to intuition, an increase in type-specific risk may lead to a decrease in diversification. Our theory is applicable to occupational choice within households, project choice under group lending, and corporate diversification.Bayesian Learning, Insurance, Risk-Sharing