Scale separation in granular packings: stress plateaus and fluctuations

It is demonstrated, by numerical simulations of a 2D assembly of polydisperse disks, that there exists a range (plateau) of coarse graining scales for which the stress tensor field in a granular solid is nearly resolution independent, thereby enabling an `objective' definition of this field. Expectedly, it is not the mere size of the the system but the (related) magnitudes of the gradients that determine the widths of the plateaus. Ensemble averaging (even over `small' ensembles) extends the widths of the plateaus to sub-particle scales. The fluctuations within the ensemble are studied as well. Both the response to homogeneous forcing and to an external compressive localized load (and gravity) are studied. Implications to small solid systems and constitutive relations are briefly discussed.Comment: 4 pages, 4 figures, RevTeX 4, Minor corrections to match the published versio

Prediction of strong shock structure using the bimodal distribution function

A modified Mott-Smith method for predicting the one-dimensional shock wave solution at very high Mach numbers is constructed by developing a system of fluid dynamic equations. The predicted shock solutions in a gas of Maxwell molecules, a hard sphere gas and in argon using the newly proposed formalism are compared with the experimental data, direct-simulation Monte Carlo (DSMC) solution and other solutions computed from some existing theories for Mach numbers M<50. In the limit of an infinitely large Mach number, the predicted shock profiles are also compared with the DSMC solution. The density, temperature and heat flux profiles calculated at different Mach numbers have been shown to have good agreement with the experimental and DSMC solutionsComment: 22 pages, 9 figures, Accepted for publication in Physical Review

Comment on the calculation of forces for multibody interatomic potentials

The system of particles interacting via multibody interatomic potential of general form is considered. Possible variants of partition of the total force acting on a single particle into pair contributions are discussed. Two definitions for the force acting between a pair of particles are compared. The forces coincide only if the particles interact via pair or embedded-atom potentials. However in literature both definitions are used in order to determine Cauchy stress tensor. A simplest example of the linear pure shear of perfect square lattice is analyzed. It is shown that, Hardy's definition for the stress tensor gives different results depending on the radius of localization function. The differences strongly depend on the way of the force definition.Comment: 9 pages, 2 figure

Stretching and folding versus cutting and shuffling: An illustrated perspective on mixing and deformations of continua

We compare and contrast two types of deformations inspired by mixing applications -- one from the mixing of fluids (stretching and folding), the other from the mixing of granular matter (cutting and shuffling). The connection between mechanics and dynamical systems is discussed in the context of the kinematics of deformation, emphasizing the equivalence between stretches and Lyapunov exponents. The stretching and folding motion exemplified by the baker's map is shown to give rise to a dynamical system with a positive Lyapunov exponent, the hallmark of chaotic mixing. On the other hand, cutting and shuffling does not stretch. When an interval exchange transformation is used as the basis for cutting and shuffling, we establish that all of the map's Lyapunov exponents are zero. Mixing, as quantified by the interfacial area per unit volume, is shown to be exponentially fast when there is stretching and folding, but linear when there is only cutting and shuffling. We also discuss how a simple computational approach can discern stretching in discrete data.Comment: REVTeX 4.1, 9 pages, 3 figures; v2 corrects some misprints. The following article appeared in the American Journal of Physics and may be found at http://ajp.aapt.org/resource/1/ajpias/v79/i4/p359_s1 . Copyright 2011 American Association of Physics Teachers. This article may be downloaded for personal use only. Any other use requires prior permission of the author and the AAP

On the 3D steady flow of a second grade fluid past an obstacle

We study steady flow of a second grade fluid past an obstacle in three space dimensions. We prove existence of solution in weighted Lebesgue spaces with anisotropic weights and thus existence of the wake region behind the obstacle. We use properties of the fundamental Oseen tensor together with results achieved in \cite{Koch} and properties of solutions to steady transport equation to get up to arbitrarily small \ep the same decay as the Oseen fundamental solution

Power-law velocity distributions in granular gases

We report a general class of steady and transient states of granular gases. We find that the kinetic theory of inelastic gases admits stationary solutions with a power-law velocity distribution, f(v) ~ v^(-sigma). The exponent sigma is found analytically and depends on the spatial dimension, the degree of inelasticity, and the homogeneity degree of the collision rate. Driven steady-states, with the same power-law tail and a cut-off can be maintained by injecting energy at a large velocity scale, which then cascades to smaller velocities where it is dissipated. Associated with these steady-states are freely cooling time-dependent states for which the cut-off decreases and the velocity distribution is self-similar.Comment: 11 pages, 9 figure

On general measures of deformation

Each particle of a continuum is assigned a second order tensor which is taken as a measure of the deformation of some neighborhood of the particle, and which is determined by a functional depending on the configurations of that neighborhood. Two invariance restrictions are imposed on the functional whose values are spatial strain tensors, that is, associated with the deformed configuration. The first requirement is that a time shift and rigid transformation of the deformed configuration leave the spatial deformation tensor unaltered relative to it. The second requires that if particles of distinct continua undergo the same deformation, the corresponding deformation tensors should be the same. For the special case in which the functional depends on the deformation in the smallest neighborhood of a particle, the restrictions imply that the deformation tensors associated with the deformed and reference configurations are isotropic functions of the left and right Cauchy-Green tensors, respectively. Jedem Teilchen eines Kontinuums wird ein Tensor zweiter Stufe als Maß für die Deformation einer gewissen Nachbarschaft dieses Teilchen zugeordnet, der durch ein Funktional bestimmt wird, das von der Konfiguration dieser Nachbarschaft abhängt. Zwei Invarianzbedingungen werden diesem Funktional, dessen Werte räumliche Verzerrungstensoren darstellen, auferlegt, und zwar im Hindblick auf die deformierte Konfiguration. Die erste Forderung besagt, daß eine Zeitverschiebung und eine starre Transformation der deformierten Konfiguration den räumlichen Verzerrungstensor im Hinblick auf diese ungeändert lassen. Die zweite Einschränkung besagt, daß entsprechende Deformationstensoren von Partikeln verschiedener Kontinua, die dieselbe Verformung erlitten haben, gleich sein sollen. Im Spezialfall, daß die Funktionale nur von der Deformation in der nächsten Umgebung des Partikels abhängen, beinhalten die Einschränkungen die Aussage, daß die mit dem deformierten und dem undeformierten Zustand verknüpften Deformationstensoren nur isotrope Funktionen des linken und des rechten Cauchy-Green Tensors sein können.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/41717/1/707_2005_Article_BF01172146.pd

Third and fourth degree collisional moments for inelastic Maxwell models

The third and fourth degree collisional moments for $d$-dimensional inelastic Maxwell models are exactly evaluated in terms of the velocity moments, with explicit expressions for the associated eigenvalues and cross coefficients as functions of the coefficient of normal restitution. The results are applied to the analysis of the time evolution of the moments (scaled with the thermal speed) in the free cooling problem. It is observed that the characteristic relaxation time toward the homogeneous cooling state decreases as the anisotropy of the corresponding moment increases. In particular, in contrast to what happens in the one-dimensional case, all the anisotropic moments of degree equal to or less than four vanish in the homogeneous cooling state for $d\geq 2$.Comment: 15 pages, 3 figures; v2: addition of two new reference

Schrodinger dynamics as a two-phase conserved flow: an alternative trajectory construction of quantum propagation

It is shown that the Schrodinger equation can be cast in the form of two coupled real conservation equations, in Euclidean spacetime in the free case and in a five-dimensional Eisenhart geometry in the presence of an external potential. This implies a novel two-phase quantum hydrodynamic model whose Lagrangian picture provides an exact scheme to calculate the time-dependent wavefunction from a continuum of deterministic trajectories where two points are linked by at most two orbits. Properties of the model are examined, including the appearance of entangled trajectories in separable states. Wavefunction constructions employing alternative two-phase models are proposed.Comment: To appear in J. Phys.

Logarithmic spin, logarithmic rate and material frame-indifferent generalized plasticity

In this work we present a new rate type formulation of large deformation generalized plasticity which is based on the consistent use of the logarithmic rate concept. For this purpose, the basic constitutive equations are initially established in a local rotationally neutralized configuration which is defined by the logarithmic spin. These are then rephrased in their spatial form, by employing some standard concepts from the tensor analysis on manifolds. Such an approach, besides being compatible with the notion of (hyper)elasticity, offers three basic advantages, namely:(i) The principle of material frame-indifference is trivially satisfied ; (ii) The structure of the infinitesimal theory remains essentially unaltered ; (iii) The formulation does not preclude anisotropic response. A general integration scheme for the computational implementation of generalized plasticity models which are based on the logarithmic rate is also discussed. The performance of the scheme is tested by two representative numerical examples