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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.

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Elastic Sheets are cool

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  1. Elastic Sheets are cool Madhav Mani

  2. 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.

  3. 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

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

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

  6. Time for some pictures

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

  8. 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

  9. 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

  10. 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 • Because during the intermediate steps, the fabric is no longer a plate, shell elements are used in the formulation

  11. In specific • Nlgeom • Largest number of maximum increments • Smallest minimum step size • Stabilization effect-dissipating energy fraction=0.00002 • Homotopy • Shell elements

  12. 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!

  13. 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!

  14. Square Geometry (shell) • Geometric effects

  15. Table Cloth • Clearly not in the regime where the instability grows

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

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

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

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