"If you, the reader, were to rip a page from this journal and crumple it, squeezing it with your hands into a
ball as hard as you can, the resulting object is still more
than 75% air. What gives this crumpled sheet its surprising strength and how does the ultimate size of the
sheet depend on the forces applied?" [emph. added] —Matan et al., "Crumpling a thin sheet"
Crumpling produces ridges—corrugations—that are both low in amplitude and wavy:
(Lin et al., "X-ray tomography of a crumpled plastoelastic thin sheet"
The phenomenon of increased flexural rigidity through corrugation allows wavy brick walls to be both thin and sturdy, with relatively little lateral instability:
Upon crumpling, the material gets in the way of itself, and substantial jumps in compressive loading are repeatedly needed to buckle new regions. (A terser description is "metastable states of regions jammed through steric hindrance.")
Contrast this behavior with that of a long smooth sheet of foil, which buckles easily because it lacks lateral support, because the critical load scales with the length squared, and because initial postbuckling can proceed by a smoothly increasing load.
than half of the interior is still filled with air. This volume
fraction is much smaller than the fcc packing, 0.74, and
the resistance divergence point, 0.75, in elastic sheets.
One question arises: How does this structure derive its incredible resistance from such an inefficient packing compared to the granular system? One reason is that the
system is constantly trapped in metastable states
due to the non-Markovian nature of the process. Another conceivable answer is from the material science. It
is known in mechanical engineering that a beam buckles under an axial load when its length exceeds 50 to
200 times of its thickness. For a shorter beam, the
deformation switches from the bending modulus to the
normally much-larger Young’s modulus. In other words,
each material has an intrinsic minimum size of facet, below which it takes an enormously large stress. We believe
this causes the poor packing of the hollow polyhedrons
surrounded by these facets inside the crumpled structure. The surprising thing is that, rather than complicating the structure, the plasticity enables the interior to reach a higher fractal dimension and more homogeneous packing, both of which contribute to its high mechanical
resistance." [emph. added] —Lin et al., "X-ray tomography of a crumpled plastoelastic thin sheet
Deboeuf et al. "A comparative study of crumpling and folding of thin sheets"
"Despite the complexity, crumpled sheets (as well as origami
structures) are comprised of only a handful of building block
structures. The four most [dominant are] the bend, the fold, the d-cone, and the stretching ridge... Slender systems are easily bent due to the tiny energetic cost of bending when compared to stretching... Folds are primarily a result of plasticity, meaning energy loss has occurred and memory has been created in the sheet (typically required for origami). Confining a sheet in three dimensions (as in crumpling a ball) leads to constraints on the sheet that can no longer be satisfied by bending
alone. Confinement forces a sheet to stretch. Stretching comes at
a high cost, so the system minimizes the total amount of
stretching present. In thin, marginally confined systems,
stretching is typically localized to a (near) singular point known
as a developable cone (d-cone). Finally, if confinement is
increased and two d-cones are created in a sheet, they are linked
by a structure known as a stretching ridge... Crumples are complex but are trustworthy materials. They are
structures dominated by bending at many different length scales
similar to a single stretching ridge." [emph. added] —Croll et al., "The compressive strength of crumpled matter"
Croll et al., "The compressive strength of crumpled matter"
Essentially, the structure formed by crumpling has some 3-D truss-like characteristics requiring substantial additional plastic deformation to obtain progressive crushing, even though much of the structure is empty air. In comparison, simple folding—even of multiple layers—requires plastic deformation only at the surfaces of each sheet, with the neutral axis (i.e., the middle of each sheet) remaining largely undeformed.
For additional discussion, references, and useful keywords, see also Lin et al.'s "Crumpling under an ambient pressure," Balankin et al.'s "Statistics of energy dissipation and stress relaxation in a crumpling network
of randomly folded aluminum foils," and Habibi et al.'s "Effect of the material properties on the crumpling
of a thin sheet."