Elastic Sheets are cool Sheets Madhav Mani

What is an elastic sheet 3-D object 3-D object Naturally flat Naturally flat Isotropic Isotropic Homogenous Homogenous Separation of scales…much thinner than it is wide Separation of scales…much thinner than it is wide Valid for pretty much anything we would refer to as a sheet…paper, clothes etc. 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 Go through some theory about large deformations of an elastic sheet…The Fopple-von Karman equations Some pretty pictures Some pretty pictures Discussion of finite element modeling Discussion of finite element modeling Results (hmm..) Results (hmm..) What now What now

Theory Why is it so hard? Why is it so hard? When is it simpler? When is it simpler? Some basic theory… Some basic theory…

Gauss Isometric transformations leave the Gaussian curvature invariant Isometric transformations leave the Gaussian curvature invariant Hence… Hence… But stretching is expensive But stretching is expensive But stretching is often localised But stretching is often localised

Time for some pictures

Some scalings Typical stretching strain: Typical stretching strain: Typical bending strain: Typical bending strain: Bending and stretching comparable: Bending and stretching comparable: Gravity length, bending and stretching due to gravity: Gravity length, bending and stretching due to gravity:

So where are the folds coming from? Energy minimization Energy minimization Gravitational energy ↓ as azimuthal angle ↓ but since inextensible folds ↑ but then energy spent in bending Gravitational energy ↓ as azimuthal angle ↓ but since inextensible folds ↑ but then energy spent in bending Hence there exists and optimal Hence there exists and optimal

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 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 Hence the optimum number of folds is

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 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 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 The loads are applied incrementally to the system, and at each step, the equilibrium equation: is solved by the Newton-Raphson method Because during the intermediate steps, the fabric is no longer a plate, shell elements are used in the formulation Because during the intermediate steps, the fabric is no longer a plate, shell elements are used in the formulation

In specific Nlgeom Nlgeom Largest number of maximum increments Largest number of maximum increments Smallest minimum step size Smallest minimum step size Stabilization effect-dissipating energy fraction= Stabilization effect-dissipating energy fraction= Homotopy Homotopy Shell elements Shell elements

Silver Lining! This project is very difficult: Non-linear, non-local, sensitive to boundary conditions This project is very difficult: Non-linear, non-local, sensitive to boundary conditions I am very glad I chose it I am very glad I chose it I am learning a lot about elasticity theory and FEM I am learning a lot about elasticity theory and FEM Who needs string theory! Who needs string theory!

Results (well…sort off) Following slides give a hint of the difficulties associated with the modeling that I have done 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! The reason I am doing this is because it’s not complete and I have no results!

Square Geometry (shell) Geometric effects Geometric effects

Table Cloth Clearly not in the regime where the instability grows Clearly not in the regime where the instability grows

Solid element (quarter cirlce) Maybe it’s working…please please work Maybe it’s working…please please work

Nope…have to use shells And it doesn’t work…but And it doesn’t work…but

So I have nothing Any suggestions? Any suggestions? Or questions? Or questions? On a positive note I conducted some experiments and the scaling laws do hold On a positive note I conducted some experiments and the scaling laws do hold