Embedding singularities via nonuniform metric
–E. Sharon,et. al. Nature 419
579 (2002); B. Audoly & A.
Boudaoud. Phys. Rev. Lett. 91, 096105 (2003).
ds02= (1 + g0(y)) dx2 + dy2
resting
distance
y
x
g0
y
flat sheet is compressed
at the boundary
stress can be relaxed by
bending, with gaussian curvature C0
C0 È ¦y ¦y g0
(Gauss embedding theorem)
actual gaussian curvature c1c2:
balance bending and stretching energy ~ g2(area)
narrow strip: single wave
suffices
b
wide strip: more waves
needed to counter increasing
g near edges
g ~ c1c2 b2
hierarchy of waves
edge of torn plastic bag