Well-posed continuum equations for granular flow with compressibility and µ(I)-rheology.

Continuum modelling of granular flow has been plagued with the issue of ill-posed dynamic equations for a long time. Equations for incompressible, two-dimensional flow based on the Coulomb friction law are ill-posed regardless of the deformation, whereas the rate-dependent µ(I)-rheology is ill-posed when the non-dimensional inertial number I is too high or too low. Here, incorporating ideas from critical-state soil mechanics, we derive conditions for well-posedness of partial differential equations that combine compressibility with I-dependent rheology. When the I-dependence comes from a specific friction coefficient µ(I), our results show that, with compressibility, the equations are well-posed for all deformation rates provided that µ(I) satisfies certain minimal, physically natural, inequalities.

Condition for alternans and stability of the 1:1 response pattern in a "memory" model of paced cardiac dynamics.

We analyze a mathematical model of paced cardiac muscle consisting of a map relating the duration of an action potential to the preceding diastolic interval as well as the preceding action potential duration, thereby containing some degree of "memory." The model displays rate-dependent restitution so that the dynamic and S1-S2 restitution curves are different, a manifestation of memory in the model. We derive a criterion for the stability of the 1:1 response pattern displayed by this model. It is found that the stability criterion depends on the slope of both the dynamic and S1-S2 restitution curves, and that the pattern can be stable even when the individual slopes are greater or less than one. We discuss the relation between the stability criterion and the slope of the constant-BCL restitution curve. The criterion can also be used to determine the bifurcation from the 1:1 response pattern to alternans. We demonstrate that the criterion can be evaluated readily in experiments using a simple pacing protocol, thus establishing a method for determining whether actual myocardium is accurately described by such a mapping model. We illustrate our results by considering a specific map recently derived from a three-current membrane model and find that the stability of the 1:1 pattern is accurately described by our criterion. In addition, a numerical experiment is performed using the three-current model to illustrate the application of the pacing protocol and the evaluation of the criterion.

Granular friction, Coulomb failure, and the fluid-solid transition for horizontally shaken granular materials.

We present the results of an extensive series of experiments, molecular dynamics simulations, and models that address horizontal shaking of a layer of granular material. The goal of this work was to better understand the transition between the "fluid" and "solid" states of granular materials. In the experiments, the material-consisting of glass spheres, smooth and rough sand-was contained in a container of rectangular cross section, and subjected to horizontal shaking of the form x=A sin(omega(t)). The base of the container was porous, so that it was possible to reduce the effective weight of the sample by means of a vertical gas flow. The acceleration of the shaking could be precisely controlled by means of an accelerometer mounted onboard the shaker, plus feedback control and lockin detection. The relevant control parameter for this system was the dimensionless acceleration, Gamma=Aomega(2)/g, where g was the acceleration of gravity. As Gamma was varied, the layer underwent a backward bifurcation between a solidlike state that was stationary in the frame of the shaker and a fluidlike state that typically consisted of a sloshing layer of maximum depth H riding on top of a solid layer. That is, with increasing Gamma, the solid state made a transition to the fluid state at Gamma(cu) and once the system was in the fluid state, a decrease in Gamma left the system in the fluidized state until Gamma reached Gamma(cd)

Directed force chain networks and stress response in static granular materials.

A theory of stress fields in two-dimensional granular materials based on directed force chain networks is presented. A general Boltzmann equation for the densities of force chains in different directions is proposed and a complete solution is obtained for a special case in which chains lie along a discrete set of directions. The analysis and results demonstrate the necessity of including nonlinear terms in the Boltzmann equation. A line of nontrivial fixed-point solutions is shown to govern the properties of large systems. In the vicinity of a generic fixed point, the response to a localized load shows a crossover from a single, centered peak at intermediate depths to two propagating peaks at large depths that broaden diffusively.

Analysis of the Fenton-Karma model through an approximation by a one-dimensional map.

The Fenton-Karma model is a simplification of complex ionic models of cardiac membrane that reproduces quantitatively many of the characteristics of heart cells; its behavior is simple enough to be understood analytically. In this paper, a map is derived that approximates the response of the Fenton-Karma model to stimulation in zero spatial dimensions. This map contains some amount of memory, describing the action potential duration as a function of the previous diastolic interval and the previous action potential duration. Results obtained from iteration of the map and numerical simulations of the Fenton-Karma model are in good agreement. In particular, the iterated map admits different types of solutions corresponding to various dynamical behavior of the cardiac cell, such as 1:1 and 2:1 patterns. (c) 2002 American Institute of Physics.

Force distribution in a scalar model for noncohesive granular material.

We study a scalar lattice model for intergrain forces in static, noncohesive, granular materials, obtaining two primary results: (i) The applied stress as a function of overall strain shows a power law dependence with a nontrivial exponent, which moreover varies with system geometry; and (ii) probability distributions for forces on individual grains appear Gaussian at all stages of compression, showing no evidence of exponential tails. With regard to both results, we identify correlations responsible for deviations from previously suggested theories.