Elastic Elastic SheetsSheets are cool are cool

Madhav ManiMadhav Mani

What is an elastic sheetWhat is an elastic sheet 3-D object3-D object Naturally flatNaturally flat Isotropic Isotropic HomogenousHomogenous Separation of scales…much thinner than Separation of scales…much thinner than

it is wideit is wide Valid for pretty much anything we would Valid for pretty much anything we would

refer to as a sheet…paper, clothes etc.refer to as a sheet…paper, clothes etc.

Outline of talkOutline of talk Go through some theory about large Go through some theory about large

deformations of an elastic sheet…deformations of an elastic sheet…The Fopple-von Karman equationsThe Fopple-von Karman equations

Some pretty picturesSome pretty pictures Discussion of finite element modelingDiscussion of finite element modeling Results (hmm..) Results (hmm..) What nowWhat now

TheoryTheory Why is it so hard?Why is it so hard? When is it simpler?When is it simpler? Some basic Some basic

theory…theory…32hdzU

z

BB

B

hdzU SS 2

)()(3 stretchinghbendinghenergy

GaussGauss Isometric transformations leave the Isometric transformations leave the

Gaussian curvature invariantGaussian curvature invariant Hence…Hence… But stretching is expensiveBut stretching is expensive But stretching is often localisedBut stretching is often localised

Time for some picturesTime for some pictures

Some scalingsSome scalings Typical stretching Typical stretching

strain:strain:

Typical bending strain:Typical bending strain:

Bending and stretching Bending and stretching comparable:comparable:

Gravity length, Gravity length, bending and stretching bending and stretching due to gravity:due to gravity:

EgREhgRh //

Rh /2/1)/( gEhRs

)1(12/

)/(23

3/1

EhB

ghBlg

So where are the folds coming So where are the folds coming from?from?

Energy minimizationEnergy minimization Gravitational energy Gravitational energy ↓ as ↓ as

azimuthal angle ↓ but since azimuthal angle ↓ but since inextensible folds↑ but then inextensible folds↑ but then energy spent in bendingenergy spent in bending

Hence there exists and Hence there exists and optimaloptimal

1/ glR

So how many folds do we So how many folds do we get?get?

So by doing the So by doing the balance of energies balance of energies above a bit more above a bit more carefully we can get carefully we can get that the optimal that the optimal wavelength wavelength

Hence the optimum Hence the optimum number of folds isnumber of folds is

4/14/3 Llg

4/13 )/( LlRn g

FEM modelingFEM modeling For For large deformationlarge deformation problem: the problem: the non-linearitynon-linearity

due to a due to a change in the geometrychange in the geometry of the body has of the body has to be considered in order to obtain a correct solutionto be considered in order to obtain a correct solution

Instead of the one-step solution found in linear Instead of the one-step solution found in linear problems, the non-linear problem is usually solved problems, the non-linear problem is usually solved iterativelyiteratively

The The loads are applied incrementallyloads are applied incrementally to the to the system, and at each step, the equilibrium equation: system, and at each step, the equilibrium equation: is solved by the Newton-Raphson methodis solved by the Newton-Raphson method

Because during the intermediate steps, the fabric is Because during the intermediate steps, the fabric is no longer a plate, no longer a plate, shell elementsshell elements are used in the are used in the formulationformulation

fqK

In specificIn specific NlgeomNlgeom Largest number of maximum incrementsLargest number of maximum increments Smallest minimum step sizeSmallest minimum step size Stabilization effect-dissipating energy Stabilization effect-dissipating energy

fraction=0.00002fraction=0.00002 HomotopyHomotopy Shell elementsShell elements

Silver Lining!Silver Lining! This project is very difficult: Non-This project is very difficult: Non-

linear, non-local, sensitive to linear, non-local, sensitive to boundary conditionsboundary conditions

I am very glad I chose itI am very glad I chose it I am learning a lot about elasticity I am learning a lot about elasticity

theory and FEMtheory and FEM Who needs string theory!Who needs string theory!

Results (well…sort off)Results (well…sort off) Following slides give a hint of the Following slides give a hint of the

difficulties associated with the difficulties associated with the modeling that I have donemodeling that I have done

The reason I am doing this is The reason I am doing this is because it’s not complete and I have because it’s not complete and I have no results!no results!

Square Geometry (shell)Square Geometry (shell) Geometric effectsGeometric effects

Table ClothTable Cloth Clearly not in the regime where the Clearly not in the regime where the

instability growsinstability grows

Solid element (quarter Solid element (quarter cirlce)cirlce)

Maybe it’s working…please please Maybe it’s working…please please workwork

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

So I have nothingSo I have nothing Any suggestions?Any suggestions? Or questions?Or questions? On a positive note I conducted some On a positive note I conducted some

experiments and the scaling laws do experiments and the scaling laws do holdhold