Thursday, February 16, 2012

1202.3160 (Peter Bella et al.)

Wrinkles as a relaxation of compressive stresses in an annular thin film    [PDF]

Peter Bella, Robert V. Kohn
It is well known that an elastic sheet loaded in tension will wrinkle and
that the length scale of the wrinkles tends to zero with vanishing thickness of
the sheet [Cerda and Mahadevan, Phys. Rev. Lett. 90, 074302 (2003)]. We give
the first mathematically rigorous analysis of such a problem. Since our methods
require an explicit understanding of the underlying (convex) relaxed problem,
we focus on the wrinkling of an annular sheet loaded in the radial direction
[Davidovitch et al., PNAS 108 (2011), no. 45]. Our main achievement is
identification of the scaling law of the minimum energy as the thickness of the
sheet tends to zero. This requires proving an upper bound and a lower bound
that scale the same way. We prove both bounds first in a simplified
Kirchhoff-Love setting and then in the nonlinear three-dimensional setting. To
obtain the optimal upper bound, we need to adjust a naive construction (one
family of wrinkles superimposed on a planar deformation) by introducing a
cascade of wrinkles. The lower bound is more subtle, since it must be
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