# Physics

A thin sheet of paper, say a currency note, droops under its own weight. But when one of its ends is held curved, it stiffens. This stiffening is shown in the adjoining figure for a sheet of card paper. How exactly does this work? An explanation in pictures follows. (Readers can reproduce these experiments for themselves with a sheet of paper, scissors and sticky tape.)

To answer this, we must first understand how flat sheets deform. An example is shown in the picture above. A flat sheet of paper hangs over a desk and droops under the weight of some wooden pieces glued to an edge. The picture also shows an identical paper sheet sliced along the length.

The basic mechanism is that of Euler bending best visualized using a view from the side (see figure above). On the left is the cross-section of a small segment of the paper sheet with thickness t. On the left, is the same segment when bent. Fibers near the top layer of this segment stretch, while those near the bottom shrink. This happens at each section, and therefore, slicing the sheet along the length essentially causes each slice to bend the same way.

When curved, the paper sheet stiffens, because the curvature distributed loads and deformations in a fundamentally different way. Slicing along the length helps visualize the deformation (see picture above). The slices splay apart, indicating that strips in the intact sheet are being stretched transversely.

Quantitatively, the bending and the stretching modulus both depend on the thickness . The bending modulus is proportional to , while the stretching modulus is proportional to . For thin objects, is much smaller than , and therefore, stretching becomes more and more prohibitive relative to bending.

This is the main influence of transverse curvature. It couples longitudinal bending to transverse stretching. And since it is much difficult to stretch paper, it does not droop either. We call this effect curvature-induced stiffening.

## Translating the principle to feet

So how do we test this principle in the foot? One way would be to load the foot under bending and compare the response of a transversely arched foot and the same foot flattened. But the foot is a complicated, multi-functional structure. It is not possible to modify just the transverse arch without affecting other parts and test the theory.

Instead, it is possible to weaken or disable curvature-induced stiffening by simply eliminating the elements that resist in-plane stretching. Slicing the along the length did it for the paper sheet. Our idea was to slice along the length of the foot and disengage any elastic tissue that resist transverse stretching to disable curvature-induced stiffening. See the Biomechanics page for the results.