  The conformation of a long linear polymer dissolved in fluid and exposed to
an extensional flow is well-known to exhibit a "coil-stretch" transition, which
for sufficiently long chains can lead to bistability. The present work reports
computations indicating that an analogous "compact-stretched" transition arises
in the dynamics of a thin elastic sheet. Sheets of nominally circular, square
or rectangular shape are simulated in planar and biaxial flows using a finite
element method for the sheet conformations and a regularized Stokeslet method
for the fluid flow. If a neo-Hookean constitutive model is used for the sheet
elasticity, the sheets will stretch without bound once a critical extension
rate, as characterized nondimensionally by a capillary number, is exceeded.
Nonlinear elasticity, represented with the Yeoh model, arrests the stretching,
leading to a highly-stretched steady state once the critical capillary number
is exceeded. For all shapes and in both planar and biaxial extension, a
parameter regime exists in which both weakly stretched (compact) and strongly
stretched states can be found, depending on initial conditions. I.e. this
parameter regime displays bistability. As in the long-chain polymer case, the
bistable behavior arises from the hydrodynamic interaction between distant
elements of the sheet, and vanishes if these interactions are artificially
screened by use of a Brinkman model for the fluid motion. While the sheets can
transiently display wrinkled shapes, all final shapes in planar and biaxial
extension are planar.
