Lecture 1

Prospectus: what does energy condensation mean?

Two similar-looking phenomena: crumpled sheets and force chains in granular material

The notion of energy condensation

Aim and plan of lectures

Crumpling: qualitative survey

Previous works related to crumpling

The energy of a deformed elastic sheet

Why large sheets are nearly unstretchable

Deforming an unstretchable sheet:

cone singularity

interaction of two cones: two cone singularities induce ridge singularity.

Stretchablity alters the ridge singularity

kite model suggests new scaling properties

Numerical search for ridge scaling properties

Properties of a ridge

"virial theorem" relating stretching bending energy
strength of ridges: modulus and buckling of a ridged structure

Properties of a crumpled sheet

ridges should dominate deformation energy
network of ridges is heterogeneous

Summary: open questions about crumpling.

Lecture 2: Differential Equation analysis

Characterizing the deformed membrane: stress and curvature potentials

Von Karmann equations for equilibrium of the membrane

Nearly flat membranes: electrostatic analogy

Lobkovsky's minimal ridge: scaling of Von Karmann equations:

with thickness
with sharpness or dihedral angle
with distance from the ridge ends
Screening length for the flanks of a ridge

numerical verification

Ridge perturbed by external force

Prediction for minimal ridge

Numerical studies of forced ridges

Interaction of two ridges
Linear response, buckling threshold:

Anticipated behavior
Minimal ridge differs from real ridges

Conclusions: many puzzles remain with forced ridges

Explaining observed scaling powers.
What "relevant" variables are needed to fix scaling behavior?

Lecture 3:

Stochastic features of crumpling

Vertex rules in a flattened sheet
Induced flattening
Ridge network in crushed membrane

Phantom, self intersecting membrane : numerical studies
Self-avoiding membrane: observations, experiments

self-avoiding membranes are heterogeneous

Predictions from scaling laws

Crumpling in general spatial dimensions

Form of elastic energy for m-dimensional membrane in d-dimensional space

Confining an elastic disk in a shrinking sphere

High-dimensional behavior: d > = 2m

No stretching; no energy condensation

Unstretchable sheet with d < 2m

Confining sphere may not shrink indefinitely

Numerical studies of 3-sheets in 4, 5, 6, dimensions

Confinement in sphere: 4 and 5 dimensions appear qualitatively different
Anticipated behavior: stack of two sheets.
m-torus in d dimensions.  d influences ridge pattern but not ridge scaling.

Lecture 4: Forces in static bead packs: hypotheses and theories       Alexei Tkachenko and TW

Lessons from crumpling: how did condensation arise?

Anecdotal evidence for heterogeneity in bead packs

Contrasting notions of stress propagation

Traditional elasto-plastic picture of force transmission
Simple unidirectional transmission: q-model
Null-stress hypothesis and isostaticity

A microscopic justification

Lecture 5: Forces in bead packs: experiments and simulations       Alexei Tkachenko and TW

Null-stress law

Spatial response to point force

Statistical distribution of forces

Lecture 6: Menagerie of energy-condensation phenomena: search for a unified view.