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One of the major thrusts of our lab is the study of granular materials. Various aspects of this work are supported by the National Science Foundation and by NASA. There are many practical situations that involve the handling of these materials. But understanding the physics necessary to describe how they flow or even their static properties is a hard problem. It is estimated that a large fraction of the US gross domestic product is spent in handling these materials. Industrial facilities for handling granular materials typically operate well below design efficiency, and catastrophic failures are all too frequent, as seen in the container collapse below.
Granular materials can exist in many different states that roughly resemble solids and fluids. If energy is not continuously provided to granular systems, they collapse into the 'solid state'. This state is characterized by strong fluctuations of the forces on individual grains. If a lot of energy is provided to these materials, they can look more fluid-like. However, analogies to ordinary solids and fluids may be misleading. In our work, we are trying to understand the basic nature of these different states of granular materials.
How are forces carried in a granular solid? Photoelastic Image shows how
How does such a system respond to a local force?
Dense granular systems are characterized by stress chains. These look like:
This false color image is taken from Dan Howell's experiments. This is a 2D experiment in which a collection of disks undergoes steady shearing. The red regions mean large local force, and the blue regions mean weak local force. The stress chains show in red. The key point is that on at least the scale of this experiment, forces in granular systems are inhomogeneous and itermittent if the system is deformed. We detect the forces by means of photoelasticity: when the grains deform, they rotate the polarization of light passing through them. We use a novel calibration of the photoelastic effect that takes advantage of the fact that the polarization vector of light can actually rotate through multiple factors of pi. You can see this in the following closeup of some of the disks. The colored bands in this image give the information on the rotation of the polarization of transmitted light.
You can read more about Dan's research at Dan Howell's web page
These experiments are joint work with Christian Veje then at CATS, part of the Neils Bohr Institute in Copenhagen, and with Hans Herrmann and Stefan Luding then at the Institute for Computer Applications at the University of Stuttgart, Germany.
An important issue for static granular materials concerns the way in which forces are carried, on average, through the material. Although this seems like a simple question that would have been settled years ago, there is considerable current debate that has been stimulated by several new models. Classical soil mechanics models assume that the material is elastic up to the point of yield, so that nondeforming materials should satisfy an elliptic partial differential equation. The q-model of Coppersmith et al. predicts a parabolic PDE, and the models of Bouchaud, Cates, Claudin and Wittmer indicate a wave equation (or a generalization), hence a hyperbolic PDE. Kenkre et al. also propose a generalized wave equation, the telegraph equation. Experiments with Eric Clement, Junfei Geng, and Guillaume Reydellet were designed to test these competing models by a Green's function measurement. We used photoelastic grains in 2D, and we observed the force response to a point perturbation. The basic process is illustrated in the figure below. We created a vertical packing of grains, and in the two lefthand images, the stress network is the result of gravity only. We then applied a point force to the upper surface, and we observed the change in the force response everywhere in the sample. The two righthand images show this kind of change. Here, (b) follows from (a) after the application of the point force, and (d) similarly follows from (c). In order to obtain mean behavior, we repeated this process many times for a given packing structure.
The image below shows a typical average response over many realizations for an ordered hexagonal packing of disks. This system has a response function where the force is carried along the principal lattice directions. This might be indicative of a wave-like response, but Goldenberg and Goldhirsch suggested that this may be described by an anisotropic elastic model.
The response in a disordered system, such as a packing of pentagonal particles, is consistent with an elastic response. The figure below shows the transition in the response function from wave-like to isotropic elastic-like as the packing changes from order. The left-most image corresponds to the highest degree of order in the packing, and the right-most image to the lowest degree of order.
We have carried other experiments in 3D using conventional granular materials such as glass beads. These experiments have focused on the stress fluctuations that occur when the stress chains are deformed. In the following two figures we show respectively the cross section of the apparatus and a time series for stress, i.e. stress vs. time during the shearing process. The apparatus consists of a relatively narrow annular channel that is filled with glass beads. We shear from above with a ring that is rough at the scale of the beads. Finally, we detect the forces at the bottom of the channel. If we replaced the glass spheres with a fluid, such as water, then we would measure nothing. But with a granular material, we observe very large stress fluctuations. In the stress time series, the bar at the lower left is the mean stress; the largest observed stresses were more than ten times greater than this mean.
Experiments in this lab are directed toward understanding the dynamics of shaken, rolling, and colliding granular materials. Granular materials rapidly condense into a dense state in the absence of continued energy input. Important parts of the "condensing" include inelastic collisions, friction, and the presence of gravity. Shaking provides a convenient way to probe these different processes. And, as soon as the shaking is strong enough to overcome gravity, shaking leads to a large number of fascinating flow states. Look below to see what some of these look like.
When granular materials are shaking up and down with large enough accelerations, they begin to convect. In our lab, Hyuk Pak, Eric van Doorn, Chris Beasley and Bob Hartley studied this system. The pictures below are some of Bob's streak photo's showing the flow lines for a two-roll state. To see more of Bob's work, look on his web page (Bob Hartley's web page)
The apparatus used for these experiments is quite simple: just a piston driven by a motor and crankshaft arrangement. It is very important to get the system very level and free of spurious vibrations, since it is unstable to any horizontal acceleration.
Under both horizontal and vertical shaking there are a collection of novel flow states. Below is a set of streak patterns from Sarath Tennakoon's work showing a slow sloshing mode that occurs when the frequencies of the vertical motion is slightly different from the frequency of the horizontal motion.
Idealized systems such as Rayleigh-Benard convection are paradigms of ideal nonlinear systems that select a pattern. Rayleigh-Benard convection is in fact just the flow that results when a perfect flat layer of fluid, such as water, is heated from below. In the "real world" such ideal systems rarely exist. We are exploring two systems "rough" systems where pattern selection still occurs, one involving Couette flow, and the other involving convection of fluid in a porous medium.
Ben Painter studied the first transition to flow in Taylor-Couette flow in which the inner cylinder is roughed by cutting a series of randomly spaced grooves. To see more on this visit Ben Painter's web page A key result from this research was the observation of a qualitatively different kind of transition to flow in which the amplitude of flow grows exponentially in the Taylor number, a dimensionless measure of the inner cylinder rotation rate. Here are some sample images of the resulting flow pattern at different rotation rates.