# Physics of ribbon curling unravelled

Scientists have explained precisely how and why a ribbon curls when we run a scissor blade down one side of it.

They teased out the effects of the blade's sharpness, the tension applied to the ribbon and the speed it moves.

As the ribbon bends around the blade, its outermost side stretches and permanently deforms, producing curls.

Sharper blades and slower movement make tighter curls - but the pulling force has an ideal strength, above which the curls become less pronounced.

The UK-based team will present the study on Wednesday at the March Meeting of the American Physical Society in Baltimore; it also appeared last month in the journal PNAS.

In their experiments, a thin ribbon - made in this case from a transparent PVC film - was draped over a blade and a weight was hung from the end. The ribbon was then wound onto a cylinder in order to drag it across the blade.

The team measured the width of curls produced by different weights and winding speeds - and also created a mathematical model to show that these could be explained by predictable changes in the structure of the ribbon.

## Irreversible curls

Senior author Anne Juel, from the University of Manchester, said it was fairly straightforward to understand why a slower movement produces greater curling:

"It takes a certain amount of time for the stress in the ribbon to relax, and the irreversible deformation to take place."

That relaxation - or "yield" - is what leaves the ribbon curled, because the outer side of the ribbon is permanently stretched compared to the side that was touching the blade.

Similarly, then, a sharper blade increases the stretch and the yield - making tighter curls.

But putting greater tension on the ribbon, with heavier weights, only increased curling up to a point.

This, Prof Juel explained, is because the deformation can spread too far into the ribbon:

"The first part that's going to start to yield is the outermost part of the ribbon, because that's the point where the stress is going to be highest. And then as you apply larger loads, the yield is going to infiltrate deeper and deeper inside the ribbon."

Eventually, with enough pulling power, the distortion of the ribbon's structure will cross the halfway point - which dampens the overall curling effect.

"So the tightest curl will be obtained when you manage to apply a load that will bring yield to exactly half the thickness of the ribbon," Prof Juel said.

And if you're wrapping a swag of presents with a few different kinds of ribbon, she added, that optimum tension will be a moving target.

"It has to be relative to the material properties of the ribbon. So it will be different for different ribbons."

Study co-author Buddhapriya Chakrabarti, of Durham University, presented some data on the same question at a previous APS meeting; Prof Juel said she and her colleagues at Manchester contacted Dr Chakrabarti when they realised they shared an interest in the problem.

Together, they have now published the first complete physical account of ribbon curling.