Magnetic Resonance Elastography (MRE) to probe the internal structure and 3D force chain network of quasi-static granular materials

MRE Development at NMR MRE is a type of MRI technique developed for medical researchers which can measure very small displacements within a elastic material undergoing a periodic mechanical oscillation. The mechanical oscillation applied to the system is phase locked to an oscillating gadient within the MRE pulse sequence, which allows the particles position at a particular point during the periodic oscillation to be encoded. It allows visualization of waves traveling through a material and also a qauntitaive measurement of the materials elastic properties. For a great overview from those who originally developed MRE the reader is referred to:R. Muthupillai et al., "Magnetic resonance elastography by direct visualization of propogating acoustic strain waves," Science, 269:1854-1857, 1995. We developed an MRE sequence based on that of Muthupillai et al. above and tested it using a room temperature gel, Perma-Gel. An MRE image taken with our sequence can be seen at right, clearly showing waves traveling through the gel.

How We Use MRE to Image Force Chains Within 3D Granular Assemblies A granular assembly is pre-stressed, so force chains are created. Then the small amplitude vibration which is phase locked to the MRE pulse sequence is applied on one side of the system. We worked at a frequency of oscillation of 400hz, therefore in the long-wavelength limit. This means that instead of seeing waves traveling through the system all the particles which are part of chains will move in unison and acquire approximately the same intensity, whereas particles which are not part of a chain will move randomly or not at all. No change is necessary to the MRE pulse sequence other than to keep the mechanical oscillation phase locked with the oscillating gradient - in other words, if you change the period of one remember to change the period of the other!

New Mexico Resonance has, therefore, made the first fully 3D images of force chains structure in three-dimensions.

Comparison of 2D and 3D Force Chains Since what we image with MRE are displacements we aren't getting information about the disrbution of forces, but rather we are acquiring the spatial distribution of the locations of the force chains. One statistic we can look at then is the distribution of the lengths of the chains. We find the distribution of chain lengths in our 3D systems to decay exponentially (left). This was seen in both 2D and 3D and for 2D systems of photoelastic disks. Furthermore, the initial conditions and forcing type of the system influence the form of the distribution (right).

This material is based upon work supported by the National Science Foundation under Grant No. 0607813.
Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.

REFERENCES:
Sanfratello, L.,Fukushima, E., and Behringer, R.P. Using MR Elastography to Image 3D Force Chains in a Quasi-Static Granular Assembly. Published in Granular Matter October 2008.
Sanfratello, L. and Fukushima, E. Distribution of Force Chain Lengths: A Useful Statistic that Characterizes Granular Assemblies. Submitted to Physical Review E, October 2008.

Last Updated: 1:03 PM 11/19/2008