Research Interests

Force Propagation in Granular Materials

Other Collaborators: Robert Hartley

Force Chains Observed in a Silo of Photo-Elastic Discs

Photo-elastic materials are special materials that have a different affect upon light passing through them depending upon the amount of force applied to the material. When photo-elastic materials are viewed through circular polarizers, applying high forces causes bright bands to show up in the material (the pattern of fringes is caused by the polarization rotating through multiple phases).

We use discs cut from photo-elastic material to study the force networks within vertically confined granular materials (for example, a grain silo). In granular materials, force is rarely transmitted uniformly, but rather preferentially along a network forming force chains -- often right next to regions where there is little or no force.

Because of these wide fluctuations of force, though objects like grain silos are common-place, their fundamental physics are still not fully understood and they are actually much more likely to fail than similar, liquid storage facilities like water towers. From an engineering standpoint, there are several systems of model equations based upon an 1895 paper by Janssen that usually work in specific situations. Understanding when and why these models fail may require understanding their basis in fundamental physics including characterizing the importance of force chains (or conversely, how the effects of force chains are minimized).

There are several models for describing the force chains observed experimentally in two-dimensional granular materials. Most models start with lattice-like descriptions of grains connected by specific contacts to argue for continuum equations of force propagation. By examining the actual force distribution in a two-dimensional hopper under controlled conditions (including the application of overloads and moving the walls hundredths of one grain diameter to change friction) we are able to test the predictions of these models.

Secondary Circulation in Granular Hopper Flows

Other Collaborators: Pierre Gremaud (Dept. of Math, NC State), Matt Matthews (Dept. of Math, UTenn-Chatanooga), Dave Schaeffer (Dept. of Math, Duke)

We tilt a hopper and observe the flow of tracer particles through a window.

Granular hoppers - storage bins that are often conical or wedge-shaped from which grains are drained by gravity -- are another commonplace system that is characterized by engineering equations applicable under specific conditions. Though equations describing the velocities and stresses of material inside hoppers have existed for decades, only recently have computational techniques allowed for the full solution of those equations in three-dimensions. These accepted equations are thought to describe the flow out of the hopper as completely radial, with each particle's trajectory being a straight line converging upon the hopper opening. Recent numerical solutions of granular hopper flow equations, however, have shown that the equations actually permit exotic, fluid-like patterns of circulation when the hopper symmetry is perturbed slightly.

We perturb a hopper by tilting the hopper slightly with respect to gravity. We are testing numerical predictions that indicate flow in non-axisymmetric hoppers (including hoppers tilted with respect to gravity) exhibits a secondary, swirling motion in addition to the previously known radial downward flow. We image flowing sand through plexiglass windows in a hopper made of brass wedges, and track the positions of dyed tracer sand grains to determine how the velocity depends upon tilt angle. We can also line the inside of the hopper with different materials corresponding to varied parameters in the simulation. By comparing the observed velocities with numerical simulations of the granular hopper flow equations we can potentially evaluate the accuracy of the equations. Using different constitutive relations - equations describing the exact relationship between positions and forces of the grains - we can look for accurate statistical descriptions of this non-equilibrium system.

Magnitude of azimuthal velocity on a spherical cap (fixed distance from opening) of hopper flow as viewed from above. The flow is symmetric about the tilt axis so only one half is given. Image credit: Matt Matthews (UTenn-Chatanooga Math)

Application of Graph Theory to Force Networks

Other Collaborators: Trush Majmudar, Brian Tighe and Josh Socolar (Dept. of Physics, Duke) and John Reif (Dept. of Computer Science, Duke)

Scalar models of the distribution of force in granular materials such as the q-model (the left-most image) capture some features but neglect fluctuations. Bond percolation models (two right-most images) capture other features more common to networks in general.

The correct mathematical description of force networks within granular materials is one of the outstanding theoretical problems that may eventually lead to an understanding of the connection between the properties of constituent grains and the granular material as a whole. One approach in which I am interested is the application of general network approaches, such as random graph theory, to characterize the statistical behavior of force networks. I have investigated force networks as subsets of the network of inter-grain contacts where only bonds carrying more than a specific fraction of the mean force are included. I have found that these networks display power-law distributions of cluster size distribution similar to bond percolation models.

I completed a computer science masters project on this subject and am currently developing a paper on the influence of anistropy.

See http://www.phy.duke.edu/~wambaugh/ForceNetworks/ for more information.

Kinematics of Three-Body Collisions

Other Collaborators: Bill Bateson (Center for Applied Scientific Computing, Lawrence Livermore), Donald Griswold (LLNL)

In an introductory physics class students are taught to analyze a collision (for example, of a ball against a wall or two cars sliding on ice) by treating the different components of the motion (the motion into the collision and the motion perpendicular) separately. This is because the equations describing the collision are linear. Recent simulations describing elastic objects as being composed of random arrangements s of nodes connected by random springs have shown that if the contact surface between two colliding objects deforms (as in bouncing on a trampoline or the crumpling of a car's hood) then perpendicular motion can be redirected. This means there might be a much larger recoil than expected and that the equations are no longer linear.

We are looking for similar phenomena in the collision of three objects simultaneously. In particular, we use three air guns to fire three ball bearings so that they collide as close to simultaneously as possible. We observe the balls before and after the collision using three high-speed (2000 frames per second) cameras and use computer algorithms to extract the velocity and rotation of the balls before and after the collision. We are hoping to observe non-linear elasticity in these three-body collisions.

Non-Linear Elasticity Effects in Spring Networks

Other Collaborators: Josh Socolar (Dept. of Physics, Duke), Dave Schaeffer (Dept. of Math, Duke)

A bond-diluted spring network under compression. Black and grey bonds indicate compressive and tensile forces respectively.

I am also interested in the application of a numerical algorithm I have developed - an iterative conjugate gradient method - to the problem of elasticity in a network of nodes connected by springs in two dimensions. This is usually considered a well-behaved system, but in fact includes non-linearities due to its dimensionality. In one dimension the equations describing two nodes can be coupled by only one spring without those springs overlapping and the system is entirely linear. In two dimensions nodes can be multiply connected, for instance three nodes in a triangular configuration. Contraction or expansion of one spring changes the positions of the two nodes connected by the spring, indirectly causing the contraction or expansion of other springs - the problem is now non-linear.

Since even a regular, two-dimensional triangular lattice of springs contains non-linearity, it is possible that the relationship between a strain, or displacement of the nodes, applied to such a network and the stress within the network is also non-linear. Further, removing random springs from a regular lattice introduces additional non-linearity whose impact upon the relationship between strain and stress might also exhibit interesting phenomena.

Because I know all the spring couplings between the nodes, I should know all of the equations governing them. If I apply the conjugate gradient method I can iteratively solve for the correct positions of each node. The non-linearity of the equations is due to the difference between the actual positions of the nodes and their positions in a perfect triangular lattice, so solving with the conjugate gradient method only produces an approximate solution. I can use that solution to recalculate the couplings and iteratively solve for the actual positions. Thus I have a method of solving non-linear equations efficiently for large systems. By carefully examining the energy of the spring network as a function of small displacements (slowly compressing the system) I can look for non-linear energy dependencies.

Determination of Granular Structure Factor

Other Collaborators: Henry Everitt (Dept. of Physics, Duke)

Over the past century scattered light of controlled wavelength (including x-rays, microwaves and lasers) have been used to probe the orientation of small particles within a bulk, including molecular structure. In particular x-rays have been used to probe liquid suspensions of neutral atoms that interact with each as if they were hard spheres. We are investigating whether it is possible to conduct similar experiments with granular materials using laser light. If we can develop such a technique, it would be possible to examine the internal behavior of three-dimensional granular materials and conduct experiments using this information.