Stress, strain rate, and free volume

Although our simulations of granular drainage using the spot model were very accurate, it is unclear how to apply more generally to other granular flows. There are two aspects to our simulation: the cooperative particle motion from the spot's influence, and the prescription for how spots move. While the former appears like a very general phenomenon in granular flow, it is unclear how to move spots in other situations, and the random walk description for granular drainage seems like a special case.

A natural direction to solve this problem would be to examine more closely the material properties of the granular flow. Granular flows support stresses, and can undergo plastic strain, so we hope to borrow ideas from plasticity theories in materials science and engineering.

Plasticity theories

Using theories based on plasticity to describe granular flow is not a new concept, and the reader should refer to Nedderman [1] and Wroth and Schofield [2] for a thorough description. Unfortunately though, if one tries to write down a continuum theory to describe a granular material, analogous to the way that the Navier–Stokes equations describe a fluid, one always runs into the problem that there is no obvious way of making a closed algebraic system: there are more unknowns than equations. This is a reflection of the fact that often granular materials exhibit a “memory” – their material properties depend on their preparation.

To make a continuum theory, researchers have had to resort to making further assumptions between the variables that describe the material. An example is the Mohr–Coulomb incipient yield hypothesis. To model a granular material, researchers make use of a quantity called mu, which is the maximal ratio of shear stress to normal stress over all possible slip directions. In order for a material to undergo plastic strain, mu must reach a critical value. To simplify the system of equations, researchers propose that in a granular flow, mu is set to value where it attains this critical value everywhere: the material is about to undergo plastic strain everywhere.

This is very strong assumption, but it has allowed researchers to make progress in solving a variety of granular flows. Indeed, by analyzing the slip lines on which the material fails, our group has created the Stochastic Flow Rule [3,4] which has allowed us to extend the spot model's validity to several other geometries. There are severe numerical problems though, since often the resulting equations can have discontinuous shocks in stress, which are frequently not observed experimentally.

Probing the behavior of a “granular element”

Nevertheless, it remains to be seen whether assumptions such as the incipient yield hypothesis are actually valid. To answer this question, I have been carrying out a detailed study into analyzing quantities such as mu, packing fraction, and strain rate in a variety of different non-homogeneous granular flows. For each flow, I have broken the simulation up into cuboidal regions several particle diameters across, and I have calculated physical quantities such as shear stress and strain rate on these cells. I view this as simulating an ensemble of microscopic “granular elements” and by analyzing correlations between the variables I can extract out information about how such an element will behave. This information can then be used to guide in creating a continuum theory or a multiscale simulation.

Simulation 1 – Drainage in a tall silo

Mu, packing fraction, and strain rate for a tall silo simulation

The above sequence of images shows mu (left), packing fraction (middle), and strain rate (right) during granular drainage from a tall silo. The color scheme for each variable is black (low), red, orange, yellow, green, blue, purple, white (high). For packing fraction, there is a striking cross-over between a high density in the top part of the container, which exhibits essentially a plug-like velocity profile, to a lower density in the region where the flow is converging on the orifice – this behavior has been seen in other situations [5]. We also see a clear correlation to strain rate, which is discussed below.

Mu, packing fraction, and strain rate in a wide silo simulation

See also a movie showing the evolution of these variables in this simulation.

Simulation 2 – Drainage in a wide silo

Another simulation was carried out looking at the same variables in a wide silo. The three images to the right show mu (top), packing fraction (middle), and strain rate (bottom).

The wide silo often exhibits very different behavior than the tall silo. There are significant stagnant regions of particles to either side of the orifice. In a tall silo, the upper part of the flow is essentially plug-like, but for the wide silo this does not happen. The distribution of stresses is also quite different. Nevertheless, we still see strong correlations between packing fraction and strain rate. Also visible is a region of higher density, high up in the central region of the flow.

See a movie showing the evolution of these variables in this simulation.

Mu, packing fraction, and strain rate in a pushing simulation

Simulation 3 – Pushing experiment

Starting from the same packing as the wide silo drainage experiment, a pushing simulation was carried out. A layer of particles at the bottom of the simulation was frozen in place, and the right half was moved upwards at a slow, constant velocity, causing the granular material to deform. This resulting deformation is very different in form to the granular drainage experiments considered above, as particles experience an external pushing, rather than falling under gravity.

The three images to the right show mu, packing fraction, and strain rate. It is clear that the majority of the deformation takes place in a widening diagonal band that originates edge of the pushing particles. Again, we see a very clear correlation between the packing fraction and the strain rate. Correlations to mu appear less clear.

See a movie showing the evolution of these variables in this simulation. Notice that although mu does not exhibit a direct correlation with strain rate or packing fraction, it tends to get large in areas before the onset of deformation.

Strain rate versus packing fraction

One can now take information from all these coarse grained granular elements, and look for cross correlations between the variables. A particularly striking correlation can be seen between packing fraction and strain rate:

Graph of strain rate versus packing fraction for three different simulations

The data points for all three simulations collapse onto the same curve. Since the three simulations are quite different in form, it gives us confidence that the correlations we see are due properties of the granular material itself, and are not artefacts of any particular granular flow. We see that once a critical strain rate is acheived, the density of the packing begins to decrease; this is consisent with ideas of shear dilation.

The above graph also suggests a possible phase transition. Points in the horizontal section of the graph correspond to particles which are essentially static. The diagonal section corresponds to regions that are undergoing flow and rearrangement.

Packing fraction versus mu

We have also used the above data to look for correlations between mu and packing fraction. From the graph, we see that values of mu can fluctuate by almost a factor of two across the simulation, which is very different from the incipient yield hypothesis, where we would expect mu to be constant. A clear correlation between mu and packing fraction cannot be seen. However, for the two drainage experiments, the points follow roughly an inverted U-shape, suggesting that mu must reach a critical value in order for a particular region to flow.

Graph of packing fraction versus mu for three different simulations

Future work

The simulations provide us with a wealth of other information to analyze. We intend to examine the validity of many other continuum hypotheses. One such is coaxiality, which says that the eigenvectors of the stress tensor and the strain rate tensor should be aligned. We also plan to look at correlations in a wider range of simulations such as shearing in a Couette cell, or flow on an inclined plane.


  1. R. M. Nedderman, Statics and Kinematics of Granular Materials, Nova Science, 1991.
  2. A. Schofield and C. Wroth, Critical State Soil Mechanics, McGraw-Hill, 1968.
  3. K. Kamrin and M. Z. Bazant, Stochastic flow rule for granular materials, Phys. Rev. E 75, 041301 (2007). [Link]
  4. K. Kamrin, C. H. Rycroft, and M. Z. Bazant, The Stochastic Flow Rule: A Multi-Scale Model for Granular Plasticity, Modelling Simul. Mater. Sci. Eng. 15, S449–S464 (2007). [Link]
  5. C. H. Rycroft, G. S. Grest, J. W. Landry, and M. Z. Bazant, Analysis of Granular Flow in a Pebble-Bed Nuclear Reactor, Phys. Rev. E 74, 021306 (2006). [Link]