Elastic Sheets are cool
Madhav Mani
What is an elastic sheet
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3-D object
Naturally flat
Isotropic
Homogenous
Separation of scales…much thinner than it
is wide
Valid for pretty much anything we would
refer to as a sheet…paper, clothes etc.
Outline of talk
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Go through some theory about large
deformations of an elastic sheet…The
Fopple-von Karman equations
Some pretty pictures
Discussion of finite element modeling
Results (hmm..)
What now
Theory
Why is it so hard?
 When is it simpler?
 Some basic theory…
 B  z
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UB 
 dz
2
B
h
3
2
U S   dz S  h
energy  h (bending )  h( stretching )
3
Gauss
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Isometric transformations leave the
Gaussian curvature invariant
Hence…
But stretching is expensive
But stretching is often localised
Time for some pictures
Some scalings
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Typical stretching strain:
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Typical bending strain:
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gRh / Eh  gR / E
h/ R
Bending and stretching
comparable:
Rs  ( Eh / g )
Gravity length, bending
and stretching due to
gravity:
l g  ( B / hg )1/ 3
1/ 2
B  Eh3 / 12(1   2 )
So where are the folds coming
from?
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Energy minimization
Gravitational energy ↓ as
azimuthal angle ↓ but since
inextensible folds↑ but then
energy spent in bending
Hence there exists and
optimal
R / l g  1
So how many folds do we get?
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So by doing the
balance of energies
above a bit more
carefully we can get
that the optimal
wavelength
Hence the optimum
number of folds is
  lg
3 / 4 1/ 4
L
n  R /(l g L)1/ 4
3
FEM modeling
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For large deformation problem: the non-linearity due
to a change in the geometry of the body has to be
considered in order to obtain a correct solution
Instead of the one-step solution found in linear problems,
the non-linear problem is usually solved iteratively
The loads are applied incrementally to the system,
and at each step, the equilibrium equation:
is solved by the Newton-Raphson method
Kq  f
Because during the intermediate steps, the fabric is no
longer a plate, shell elements are used in the
formulation
In specific
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Nlgeom
Largest number of maximum increments
Smallest minimum step size
Stabilization effect-dissipating energy
fraction=0.00002
Homotopy
Shell elements
Silver Lining!
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This project is very difficult: Non-linear,
non-local, sensitive to boundary conditions
I am very glad I chose it
I am learning a lot about elasticity theory
and FEM
Who needs string theory!
Results (well…sort off)
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Following slides give a hint of the
difficulties associated with the modeling
that I have done
The reason I am doing this is because it’s
not complete and I have no results!
Square Geometry (shell)
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Geometric effects
Table Cloth
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Clearly not in the regime where the
instability grows
Solid element (quarter cirlce)
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Maybe it’s working…please please work
Nope…have to use shells
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And it doesn’t work…but
So I have nothing
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Any suggestions?
Or questions?
On a positive note I conducted some
experiments and the scaling laws do hold