@article{THESIS,
      recid = {6449},
      author = {Pal, Anshuman Sankar},
      title = {Faceted Wrinkling at a Contracted Curved Boundary:  Isometry and Hysteretic Wavelength Selection},
      publisher = {University of Chicago},
      school = {Ph.D.},
      address = {2023-06},
      number = {THESIS},
      abstract = {Thin elastic sheets buckle to form a wide variety of  morphologies that can be broadly categorised intotwo  groups: smoothly wrinkled shapes, and sharply faceted  shapes. According to common knowledge, the
former is  energetically dominated by external sources of work (like a  substrate or boundary tension), while
the latter is  dominated by sources of work arising from the sheet’s  intrinsic elasticity and geometry. In this
thesis, we  analyse a buckled morphology that shows characteristics of  both these aforementioned categories.
We call this  intermediate category “faceted wrinkling”. Using numerical  finite-element simulations, we study
a minimal  two-dimensional system: a circular annulus contracted at  the inner boundary by fraction ∆, so
that it buckles into a  radial wrinkling pattern that shows sharp zig-zag faceting  at the inner boundary. In
our first result, we argue that  this morphology results from the fact that the wrinkling is  asymptotically
isometric, i.e. its stretching energy  approaches zero relative to its bending energy. To this  end, we compare
our numerically generated solutions to an  Ansatz zero-thickness solution made up of alternating  triangles
and cones that is developable, and hence  isometric, by design. We find this isometric cone-triangle  Ansatz to
agree with simulations over a wide range of  values of system size, thickness, and wrinkle wavenumber  and
amplitude. In our second result, we address the  mechanism that selects the wrinkle wavelength λ in such  a
pure-bending configuration. Usually, wavelength selection  in elastic wrinkling occurs through macroscopic
competition  between the sheet’s bending energy and some external source  of deformation work, like a (real
or effective) substrate.  What could select λ in the absence of any competition to  the bending energy? Using
our numerical simulations, we  argue that competition between stretching and bending  energies at mesoscopic
scales leads to the selection of a  wavelength scale sensitive to both the width w and  thickness t of the sheet:
λ∗ ∼ w2/3t1/3∆−1/6. This scale λ∗  corresponds to an arrest criterion for wrinkle coarsening  in the sheet
starting from any wavelength finer than λ∗: λ  ≲ λ∗. However, the sheet can support coarser wavelengths,
λ  ≳ λ∗, since there is no penalty to their existence. Since  this wavelength selection mechanism depends on
the initial  λ, it is path-dependent (or hysteretic).},
      url = {http://knowledge.uchicago.edu/record/6449},
      doi = {https://doi.org/10.6082/uchicago.6449},
}