## Modeling of dense granular materials

Granular materials are common in everyday experience and are of critical importance in many industrial processes. Their physical basis remains poorly understood due to many complex effects, such as a combination of liquid-like and solid-like behavior, and stress inhomogeneities on the microscale. In my work I have made extensive use of the Discrete-Element Method (DEM), which is the technique of choice for simulating many industrial granular flows, whereby individual particles are modeled according to Newton's laws. However, since the contact models in DEM are based on stiff differential equations, a very small time-step is required, meaning that even relatively small problems require weeks of time on a parallel computer.

## Analysis of granular flow in a pebble-bed reactor

In collaboration with Gary Grest and James Landry at Sandia National Laboratories, I worked on analyzing large-scale simulations of an experimental pebble-bed nuclear reactor design, featuring 440,000 spherical particles of diameter 6 cm that are slowly cycled through a cylindrical container. Some designs feature two types of particles (fuel and moderator) and a major focus of the study was to analyze the amount of particle mixing that occurs during flow. The DEM simulations in this study required over a month of time on 90 processors, since the contact models are stiff and a small timestep is required. In many situations, such as in real-time control or in optimization this would be infeasible, and there is a need to develop rapid simulation methods, particularly those that can correctly capture particle mixing, which provided a motivation for my subsequent work.

#### References

- 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]

## A stochastic model of mixing in dense granular drainage

Snapshots of granular drainage using discrete-element simulation (left) and spot simulation (right).

I worked on developing the “spot model” with Martin Bazant, as a multiscale simulation method for modeling flow and mixing in dense granular drainage. Particles in a granular material form a random packing, and a single particle is strongly geometrically constrained by its neighbors, so in order to flow it must move co-operatively with them. In the spot model, this behavior is captured by breaking down the flow into correlated motions of small groups of particles, on a mesoscopic length scale of three to five particle diameters. With a few physically-justified fitting parameters, I was able to accurately reproduce many features of a discrete-element simulation of granular drainage from a hopper, preserving the packing constraints between particles during the flow. Since the algorithm does not need to calculate the complex force networks between particles, but concentrates on the geometry, it requires the approximately a hundredth of the computational cost of DEM. This approach provides a rapid way to simulate granular materials flowing through hoppers that would be appropriate in certain industrial situations where mixing would need to be estimated in real time. Recently, I have written an additional paper that discusses many practical aspects of simulations of this type, such as free surface and boundary layer modeling.

#### References

- C. H. Rycroft, M. Z. Bazant, J. W. Landry, and G. S. Grest,
*Dynamics of Random Packings in Granular Flow*, Phys. Rev. E**73**, 051306 (2006). [Link] - C. H. Rycroft, Y. Wong, and M. Z. Bazant,
*Fast spot-based multiscale simulations of granular flow*, Powder Technol.**200**, 1–11 (2010). [Link]

## Studies of granular rheology

Maps of normalized shear stress, packing fraction, and deformation rate computed from a DEM simulation

The spot model only applies to granular drainage, although some subsequent analytical work suggested a connection to the Mohr–Coulomb plasticity model. Using this it was possible to extend the spot model to flow in a Couette shear cell. However, some of the continuum assumptions behind Mohr–Coulomb model are questionable, and even in simple situations it can predict shocks in stress that are not observed in experiments. To test these assumptions, I carried out a study to examine changes in physical quantities such as stress, strain rate, and packing fraction in a variety of non-steady, non-homogeneous, dense granular flows. At the level of a single particle, it is difficult to define these quantities, and discrete effects prohibit a good continuum interpretation. However, it is possible to approximately define these quantities at the mesoscopic scale. The flows can be broken down into boxes on this length scale, in which the physical quantities can be computed. By performing statistical analyses, we are able to test the continuum assumptions in the Mohr–Coulomb theory. The technique provides a general method for testing and developing continuum models in many types of amorphous material.

#### References

- 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] - K. Kamrin and M. Z. Bazant,
*Stochastic flow rule for granular materials*, Phys. Rev. E**75**, 041301 (2007). [Link] - C. H. Rycroft, K. Kamrin, and M. Z. Bazant,
*Assessing continuum relationships in simulations of granular flow*, J. Mech. Phys. Solids**57**, 828–839 (2009). [Link]

## Testing the accuracy of discrete-element simulation

A general question with the use of discrete-element simulation is to understand how accurately it matches real experiment. I worked with Arshad Kudrolli and Ashish Orpe to carry out a quantitative, three-dimensional microscopic comparison between DEM simulation and a laboratory granular flow in order to provide a better understanding of the limitations of simulation. We chose to examine a granular shearing experiment, whereby flowing grains (shown in blue) move past a rough wall made of glued grains (shown in yellow).

The laboratory experiments are carried out by using glass beads that are immersed in a fluid. By precisely matching the index of refraction of the fluid and the beads, it is possible to image into the three-dimensional bulk of the flow by making use of a laser sheet. The simulations were carried out using exactly the same geometry as the experiments. The simulations match the many features of the experiments (such as grain velocities and grain ordering) to a high degree of accuracy, although there are some discrepancies that arise from limitations in the contact force model between grains.

#### References

- C. H. Rycroft, A. V. Orpe, and A. Kudrolli,
*Physical test of a particle simulation model in a sheared granular system*, Phys. Rev. E.**80**, 031305 (2009). [Link]

## Analysis of graphite wear and scaling properties in a pebble-bed reactor

In recent work, I collaborated with nuclear engineers at the Paul Scherrer Institute in Switzerland, on the design of an experimental facility to examine graphite dust buildup in pebble-bed nuclear reactors due to rubbing between pebbles, which has been identified as a key safety issue in early prototypes. Due to cost limitations, the PSI was considering building a scaled-down facility, but this motivated a general question of scientific and engineering importance: how do granular flows scale? To address this, I carried out discrete-element simulations in a full-size reactor geometry, and compared them to flows scaled down by factors of 3:1 and 6:1. In the images below, the pebbles are colored according to the amount of wear that they experience.

The simulations were used to investigate how many different features of flow vary at the different scales. The results show that the scaling of granular flows is difficult, and there is evidence of complex behavior due to discrete pebble packing effects, although several simple scaling rules can be derived.

#### References

- C. H. Rycroft, A. Dehbi, T. Lind, and S. Güntay,
*Granular flow in pebble-bed nuclear reactors: Scaling, dust generation, and stress*, Nucl. Eng. Design.**265**, 69–84 (2013). [Preprint] - T. Lind, S. Güntay, A. Dehbi, Y. Liao, and C. H. Rycroft,
*PSI Project on HTR Dust Generation and Transport*, proceedings of HTR 2010, Prague, 2010. - C. H. Rycroft, T. Lind, S. Güntay, and A. Dehbi,
*Granular flows in pebble bed reactors: dust generation and scaling*, proceedings of ICAPP 2012, Chicago, June 24–28, 2012. [Paper]