**Scott Victor Franklin**, Rochester Institute of Technology

The primary accomplishment in the past year was adapting a Weibullian
"weakest link" statistical theory to explain our experimental results.
A collaboration was initiated with Steve Teitel at the University of
Rochester to expand our computational abilities. During my sabbatical
year (2013-2014), I will focus almost exclusively on implementing a
fully parallelized code for three-dimensional, frictional particles.

*Weibullian "weakest link" statistics*

In our experiments, a cylindrical pile of U-shaped staples is
subjected to a linear extentional stress. We detect when the applied
force drops, indicating pile elongation, and the statistics of these
events analyzed. Previously, our most intriguing finding was that
longer piles are significantly weaker than short piles (as described
in our 2012 report). This can now be explained through a Weibullian
weakest link theory as follows.

The theory assumes that long samples are comprised of *N* smaller
units, each with a length *δL*, and the sample fails
(breaks) if any one of the sub-units fails. The theory further
assumes that the probability of failure is proportional to the
length *δL* and scales with the applied force *F* as a
power law with exponent *m*. The total probability *S* that
a sample of total length *L* does not fail is the product
of *N* individual probabilities for
success: *S*=∏[1-*F ^{m}δL*]. Taking the
logarithm of each side and keeping terms to highest order results in
the approximation that

If the pile length is held constant, the theory predicts a failure
probability that goes as 1-exp[*-F ^{m}*]. A plot of
experimental data at fixed

With the scaling exponent from the fixed-length measurements, we may
now make two predictions to further test the theory's applicability.
The first is the mean yield force as a function of sample length.
This is obtained by integrating the two-dimensional probability
distribution over all lengths:
<*F _{Y}>=∫ LF* exp[

We emphasize again that a measurement taken at a *single length*
has now been used to predict mean failure probabilities *at multiple
lengths*.

Finally, the entire corpus of data, consisting of over 7500 individual
slip events, may be collapsed onto a single master curve. The
functional form of the master curve is a bit unwieldy, involving the
normalized 2-d probability distribution function, but in simplest form
it predicts
that *L ^{-1}*log[1-

Subsequent work will disentangle the statistics of events that occur before and after the single largest rearrangement, testing the assumption that all events may be treated as independent. The Weibullian theory also makes predictions about how yield forces should scale with changes in cross-sectional area and also in response to bending forces.

*Computational work*

A collaboration with Steve Teitel at the University of Rochester was
initiated. Steve's previous work has involved simulations of 2-d
round, frictionless particles under shear. We have developed code,
parallelized to run on a GPU, for 2-d U-shaped particles under shear
and are investigating the shear modulus and response of the pile to
various imposed strain-rates.

For my upcoming sabbatical year, I will be working at the University of Rochester to extend these simulations to 3-d samples in both shear and extensional geometries.

Copyright © 2014 American Chemical Society