Elastic Sheets are cool
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Transcript Elastic Sheets are cool
Elastic Sheets are cool
Madhav Mani
What is an elastic sheet
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
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
UB
dz
2
h
B
3
2
U S dz S h
energy h (bending) h(stretching)
3
Gauss
Isometric transformations leave the
Gaussian curvature invariant
Hence…
But stretching is expensive
But stretching is often localised
Time for some pictures
Some scalings
Typical stretching strain:
Typical bending strain:
gRh/ Eh gR / E
h/R
Bending and stretching
comparable:
Rs (Eh / g )
Gravity length, bending
and stretching due to
gravity:
l g ( B / hg )1/ 3
1/ 2
B Eh3 / 12(1 2 )
So where are the folds coming
from?
Energy minimization
Gravitational energy ↓ as
azimuthal angle ↓ but since
inextensible folds↑ but then
energy spent in bending
Hence there exists and
optimal
R / lg 1
So how many folds do we get?
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 /(lg L)1/ 4
3
FEM modeling
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
Kq f
Because during the intermediate steps, the fabric is no
longer a plate, shell elements are used in the
formulation
In specific
Nlgeom
Largest number of maximum increments
Smallest minimum step size
Stabilization effect-dissipating energy
fraction=0.00002
Homotopy
Shell elements
Silver Lining!
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)
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)
Geometric effects
Table Cloth
Clearly not in the regime where the
instability grows
Solid element (quarter cirlce)
Maybe it’s working…please please work
Nope…have to use shells
And it doesn’t work…but
So I have nothing
Any suggestions?
Or questions?
On a positive note I conducted some
experiments and the scaling laws do hold