The threshold of asymptopia
How thin is thin enough?
Both I and Mahadevan have been showing many pictures of crumpled sheets
of fabric, paper, etc. I said that if a sheet is thin enough, you
can see certain asymptotic scaling properties. But how thin is thin
enough, and what does an asymptotically thin sheet look like. To
answer this question I asked my student Brian Didonna to make a sheet that
was close to having asymptotic behavior. This sheet has a ratio of
bending to stretching energies of 5.11 to 1. That is two percent
higher than the asymptotic value of 5. Here's what the sheet looks
like, when it is formed into a tetrahedron. You can see that the
radius of curvature across the ridge is more than ten times smaller than
the length of the ridges.
Brian's simulation has 79 lattice spacings between two vertices.
The elastic thickness is .00316 lattice spacings. the gray scale
is proportional to stretching energy density to the 1/6 power.
