| Slide 1 |
| Last time... |
| Lecture 2: more singularities and ways to understand them |
| using gaussian curvature ÒchargeÓ to estimate stretching energy | ||
| vanishing mean curvature puzzle | ||
| Foppl von Karmann formalism to determine sheet shape | ||
| Lobkovsky scaling: strong constraints on ridge shape | ||
| Strength and heterogeneity of crumpled sheets | ||
| deformation from within: embedding singularities from nonuniform metric | ||
| Estimating stretching energy from gaussian curvature |
| Numerical representation: Seung-Nelson lattice |
| d-cone from pushing sheet
into container: rim structure |
| Curvature cancellation is robust |
| Experiment shows curvature cancellation |
| Distant changes alter curvature cancellation |
| damage: remove bending energy from core region | ||
| central force F decreases | ||
| curvature cancellation is incomplete | ||
| \cancellation † uniform elasticity | ||
| Known forces constrain rim curvature |
| What principle underlies
the vanishing curvature? Curvature cancellation conjecture |
| (When an elastic flat sheet of elastic thickness h is constrained by normal forces that are unbalanced except over distances R >> h, the shape converges pointwise as h/R¨0 to an a developable surface with an uncurved director through every point. For certain choices of constraint forces, the directors converge to a point, thus forming a developable cone.) | |
| If part of a d-cone is constrained (with nonzero normal force) to follow a curve that imposes nonzero curvature perpendicular to the director lines, then the Àmean curvature averaged over this curve vanishes when h/R¨0 ? . |
| Foppl van Karmann equations |
| reveal new scaling properties | |
| Possible clue to vanishing mean curvature puzzle. |
| Foppl von Karmann equations dictate the equilibrium shape |
| Scaling of minimal ridge |
| Lobkovsky scaling prediction: strong focusing near vertex |
| Embedding singularities via nonuniform metric |
| The crumpled state: how strong? how heterogeneous? |
| Buckling ridges cause crumpling noise |
| E. Kramer, | |
| A. Lobkovsky | |
| 1996 |
| Buckling cascade accounts for broad crumpling noise |
| How strong are real crumpled sheets? Mylar experiment |
| Open questions about the crumpled state |
| Organization of the ridge network: | ||
| Not random. vertices force each other | ||
| characteristic motifs appear. | ||
| How chaotic? if tiny external load is cycled, the ridge network must return to its initial state. How does this reversibility degrade as the amplitude increases? | ||
| How controllable? Can we control the pressure-volume relation by engineering the material? | ||
| Conclusions about embedding singularities |
| Slide 23 |
| Gaussian charge optimisation confirms ridge scaling |
| Stretching energy can be expressed as an integral of gaussian curvature analogous to electrostatic energy. | |
| A stretching ridge has negative gaussian curvature. | |
| It must be compensated by an equal amount of positive gaussian curvature on the adjacent flanks | |
| Optimising the width w of this ÒchargeÓ distribution gives | |
| w ~ X (X/h)-1/3 , confirming other methods. |
| Slide 25 |